Miller Analogies Test (MAT)

Preparing to take the Miller Analogies Test (MAT)?


You’ve found the right page. We will answer every question you have and tell you exactly what you need to study to pass the MAT.

Quick Facts

Get the “need to know” information at a quick glance.


The Miller’s Analogies Test is taken by students who are seeking admittance to graduate school programs. The test, which contains 120 analogy questions, must be completed within 1 hour.


Let’s look at an overview of what you can expect on test day.


The cost of the Miller Analogies Test varies by testing location. Check with your local testing center for the exact cost of the exam. The typical exam fee is $70 – $100. If you are unsure of how to register for the MAT exam, contact your local testing center.

The cost includes sending your score to up to three schools. It will cost an extra $25.00 per school if you would like your score sent to additional schools.


Here’s the MAT score breakdown:

MAT scores range from 200 to 600 and the average MAT score is about 400.

The percentile ranks included on your Official Score Report will indicate the percentage of recent test-takers who received scores lower than yours. Percentile ranks range from 1 to 99, with 99 being the best.

Immediately after you complete the test, you will see a “No Score Option” on your screen. If you choose this option, your score will not be sent to any of the schools you chose. Your testing fee will not be refunded if you choose this option, and you will not receive a Preliminary Score Report.

Pass rate

There is no passing score for this exam. Graduate admissions offices consider MAT scores as well as MAT percentile rankings, but they also consider other factors, such as grade point average.

Study Time

The time which you will need to spend preparing for the exam depends upon your content knowledge, your ability to understand analogies, and your general comfort level in testing situations.

Most students need to spend about 1-2 months studying for the MAT. You can use the resources on our site to determine your areas of weakness and spend the majority of your study time focusing on those areas.

What test takers wish they would’ve known:

  • Theoretically, you may take the MAT as frequently as you would like, but beware of retaking the exam! Many test-takers are surprised to learn that schools can see how many times you have retaken the MAT and if you increase your score by more than 50 points within 12 months, your scores are automatically cancelled. It is better to score well on the exam the first time.
  • Being a reader will improve your chances of scoring well. This habit means that you will frequently consider words and their relationships and you will be exposed to an array of concepts. If this is not already a habit of yours, get into a habit of reading as soon as you can!
  • Since test performance depends greatly on language concepts, non-native English speakers may especially benefit from additional study time. 
  • It’s important to assess your weaknesses before creating a study plan. Otherwise, you may spend too much time focusing on the wrong concepts.
  • Study a little bit each day. Many students who try to cram before the exam regret the choice.
  • Answer every question on the exam. You are not penalized for incorrect responses. It is better to guess than to leave a choice blank. If you are able to, eliminate some answer choices before you guess.
  • Do not make any plans the night before your test. Take the time to relax and get plenty of sleep. Make sure that your identification, directions to the testing center, and any other materials that you need are together the night before the exam. You don’t want to be hurried and distressed right before you take the test!
  • Eat a good breakfast before the test.
  • Plan to arrive at the testing center a few minutes early.
  • Wear layers. You can remove or put on a light jacket or sweater in order to be more comfortable in the testing center, which could be warm or cool

Test details obtained from the Pearson Assessments site. Pearson Assessments is the creator and owner of the MAT.

Content Objectives

The MAT has 120 questions. You will have 60 minutes to complete this content area.

The analogies on the test can be neatly divided into five content objectives:

  • Words and Language
  • Humanities
  • Natural Sciences
  • Social Sciences
  • Mathematics

So, let’s talk about them.

Words and Language

This content objective is designed to test your knowledge of English grammar and language by asking you to recognize the relationships between words.

Let’s look at some concepts for this content objective.


Denotation is the dictionary definition of a word. In other words, it is what a word literally means. It’s a good idea to brush up on your vocabulary before the test, but how can you remember so many new words?

The answer is to use mnemonic devices. A mnemonic device is a memory technique which involves association. Let’s look at some examples of how to use mnemonic devices to remember words.

These examples focus on using rhymes, but there are other ways to make mnemonic devices work for you – any association will do!


Now you know that a word’s denotation is what it literally means. A word’s connotation refers to the thoughts, feelings, and ideas associated with a word. 

For example, skinny has a negative connotation. Thin can mean the same thing as skinny, but its connotation is more positive.

Review the examples below.

Try completing the following analogy:

Miserly : Frugal :: Boisterous : a. Inexpensive b. Rowdy c. Energetic d. Euphemistic

This analogy tests your knowledge of denotation and connotation. Miserly and frugal mean nearly the same thing, but miserly has a more negative connotation. The word which means nearly the same thing as boisterous, but has a more positive connotation is energetic. C is the correct answer. While rowdy is also similar in meaning, rowdy has a negative connotation.


On the MAT, you are likely to be presented with analogies that test your knowledge of grammar. Let’s review a few terms and concepts associated with grammar.

Try completing the following analogy:

Adjective : Noun :: Adverb : a. Pronoun b. Verb c. Antecedent d. Phrase

An adjective describes a noun. An adverb describes a verb. The correct answer is B.

Word Parts

Being able to identify word parts is a very important skill for students who take the MAT. Sometimes, you will be given analogies in which you must identify a word part. However, you can benefit from your knowledge of word parts on almost every analogy on the exam.

Why are word parts so important?

Words can be reduced to their word parts. Every question on the exam tests your knowledge of words in some way. If you do not understand what a word means, you cannot complete an analogy correctly unless, of course, you happen to make a lucky guess.

If you understand the meaning of common word parts, you can use that knowledge to make meaning of unfamiliar words on the exam. 

Review the chart below. The example word parts may be very helpful to know on the actual exam.

Try completing this analogy:

Gaseous : Aero- :: Aqueous : a. Des- b. Emb- c. Osteo- d. Hydro-

Aqueous means “having the quality of water” and gaseous means “having the qualities of gas.” “Aero-” is a root word that means air, which is gaseous. So, which answer is a root word which refers to an aqueous substance? D, Hydro- is correct.


MAT questions regarding pronunciation test your knowledge of how to correctly say words. Try the following example:

Worcester : Versailles :: Gloucester : a. Illinois b. Connecticut d. Montenegro d. Jalisco

This analogy refers to places, but it is not asking for your knowledge of geography. Worcester and Gloucester are rhyming words; they have a similar pronunciation. The correct answer to the analogy is Illinois

Why? Both Versailles and Illinois end with a silent “s.”

Try another analogy:

Tangelo : Acaí :: Window : a. Façade b. Foyer c. Chimney d. Balustrade

The first two words are types of fruit. A window is part of an architectural feature of a home, but how do you choose the correct word?

The answer, again, is pronunciation. Tangelo rhymes with window and acaí rhymes with chimney.

If you aren’t able to find the answer from the meanings of the words alone, consider how they are pronounced!


Questions relating to the Humanities Content Objective on the MAT usually relate to the arts, religions, history, philosophy, and languages. We’ll explore some of these concepts now.

Art History

Let’s take a look at art history, a category that often appears as an MAT humanities analogy. If you don’t know Michelangelo from Dalí, don’t worry–we’ll review these artists and others right now.

Try to commit some of the following table to memory; you will be asked a follow-up question.

Now, try this analogy:

Munch : Norway :: Matisse : a. France b. Spain c. Denmark d. Painter

Munch was a Norwegian painter and Matisse was a French painter. The correct choice is A.

So, will arts that are not visual appear on the MAT? Often, other arts make an appearance as well. The image below provides a timeline of major artistic periods in history as well as various types of artists associated with those periods.

Comparative Religion and Mythology

Versions of the MAT often include some analogies about religions or mythology. You are less likely to see these types of analogies than other Humanities analogies though, so let’s just cover the basics.

Use the following chart to review some concepts and terms associated with religion.

Now that we’ve compared some of today’s religions, let’s take a look at ancient mythology. While many cultures have their own mythologies, you are most likely to encounter Greek gods and their Roman counterparts:

Now, try this analogy without referring to the table:

Romance: Aphrodite :: War : a. Mars b. Ares c. Artemis d. Neptune

In ancient Greece, romance was associated with Aphrodite. So, who’s associated with war? Both Ares and Mars, but Mars was a Roman god, so B is the correct answer.


So, what do you need to know about literature to perform well on the Humanities analogies? Well, you might be asked about a type of literature, a broad literary term, or a specific author.

Let’s review some basic literary terms:

  • Antihero: A flawed, unlikely main character
  • Alliteration: Repetition of consonant sounds
  • Assonance: Repetition of vowel sounds 
  • Ballad: Poem that is meant to be sung, tells a story
  • Bildungsroman: “Coming of age” novel
  • Climax: Point at which the action of a work peaks
  • Couplet: Two consecutive lines that rhyme with each other 
  • Denouement: End of a work, when the action winds down
  • Epic: Long heroic poem 
  • Hyperbole: Extreme exaggeration 
  • Idiom: Saying with a figurative meaning 
  • Imagery: Language that appeals to the five senses
  • Metaphor: Comparison that does not use “like” or “as”
  • Onomatopoeia: Words that imitate sounds
  • Personification: Human attributes are given to a non-human
  • Plot: Events of a work 
  • Prose: A work which is not poetry
  • Simile: Comparison using “like” or “as”
  • Sonnet: Poem of 14 lines 
  • Stanza: Section of a poem; formatted like a paragraph

Try out this analogy:

Sonnet : Poetry :: Bildungsroman a. Couplet b. Stanza c. Autobiography d. Prose

A sonnet is a type of poetry. A bildungsroman is an example of prose. D is the correct answer.

Now, think about all of the famous authors, poets, and literary works created over centuries–that’s a lot of literature! That’s also one reason why readers tend to have an advantage over non-readers on the MAT.

Let’s review a few of the most renowned writers. The list below includes poets and authors. One of the most famous works by each author is listed by his or her name.

  • Alcott, Louisa May: Little Women 
  • Alighieri, Dante: The Divine Comedy
  • Angelou, Maya: The Color Purple 
  • Austen, Jane: Pride and Prejudice 
  • Brontë, Charlotte: Jane Eyre 
  • Brontë, Emily: Wuthering Heights 
  • Caroll, Lewis: Alice in Wonderland 
  • Chaucer, Geoffrey: The Canterbury Tales 
  • Defoe, Daniel: Robinson Crusoe 
  • Dickens, Charles: Oliver Twist
  • Hawthorne, Nathaniel: The Scarlet Letter 
  • Hemingway, Ernest: The Old Man and the Sea 
  • Homer: The Odyssey 
  • Hugo, Victor: Les Misérables 
  • Kafka, Franz: The Metamorphosis 
  • Kipling, Rudyard: The Jungle Book
  • Melville, Herman: Moby Dick Miller
  • Milton, John: Paradise Lost 
  • Morrison, Toni: Beloved 
  • Nabokov, Vladimir: Lolita 
  • Orwell, George: Animal Farm 
  • Plath, Sylvia: The Bell Jar 
  • Poe, Edgar Allan: “The Raven” 
  • Seuss, Dr.: The Cat in the Hat 
  • Shelley, Mary: Frankenstein 
  • Sinclair, Upton: The Jungle 
  • Stevenson, Robert Louis: Treasure Island 
  • Stowe, Harriet Beecher: Uncle Tom’s Cabin 
  • Swift, Jonathan: Gulliver’s Travels 
  • Thoreau, Henry David: Walden 
  • Tolkien, J. R. R.: The Lord of the Rings 
  • Tolstoy, Leo: Anna Karenina 
  • Twain, Mark: The Adventures of Huckleberry Finn

Whitman, Walt: Leaves of Grass

Natural Sciences

Now that we’ve reviewed some Humanities Content Objectives, let’s take a look at the Natural Sciences. We’ll start with astronomy.


Astronomy is a branch of science which focuses on space, celestial bodies, and the universe. Here are some key terms to know:

  • Asteroid: A body of metal and rock that orbits the sun
  • Black Hole: A body of mass so dense that nothing, including not light, can escape once it has entered its gravitational field
  • Comet: A mass of ice and rocky debris that orbits the Sun in an ellipse
  • Constellation: A distinctive grouping of stars
  • Earthshine: Sunlight reflected by Earth that makes Earth’s moon “glow” 
  • Eclipse: When the shadow of a planet or moon falls upon another celestial body
  • Galaxy: A group of stars, gas, and dust that is about 10,000-100,000 light-years in diameter
  • Light-year: The distance that light travels in a year, approximately 6 trillion miles
  • Magnitude: Describes the brightness of a star or other celestial object
  • Meteor: The visible path of a meteoroid in the atmosphere 
  • Meteorite: A meteoroid that is at least partially intact after hitting a celestial body
  • Meteoroid: Smaller pieces which have broken off from asteroids.
  • Orbit: The curved path of an object around another object 
  • Sunspot: Dark areas on the sun’s surface which are relatively cool.

Next, we’ll review biology.


In case you haven’t taken a biology course lately, it might be useful to remember that biology is the study of living organisms. The word biology means the study of life. The suffix –ology means “the study of,” and -ologies, including those that are related to biology, and those that are not, are very frequently included on versions of the MAT.

Here are a few other -ologies which are encompassed by biology.

Ready to test your knowledge? Try completing this analogy: 

Algology : Phycology :: Malacology : a. Zoology b. Mycology c. Ornithology d. Conchology

Algology and phycology are synonyms; they both refer to the study of algae. Malacology and conchology both refer to the study of mollusks, so D is correct. How can you remember conchology? 

Easy, use a mnemonic device – think of a conch, which is a mollusk!

Now that you’ve reviewed various biological pursuits, let’s zoom in with a microscope. The following terms are helpful to remember when thinking of the building blocks of life:

  • Cell membrane: Acts as a wall between a cell the outside environment
  • Chloroplast: Plant organelles in which photosynthesis occurs
  • Chromosome: Made of DNAs, contains genes
  • Cytoplasm: The liquid inside of a cell  which surrounds organelles
  • DNA: Deoxyribonucleic acid, contains genetic information
  • Gene: A unit of heredity
  • Meiosis: Cell division process in sexual reproduction
  • Mitosis: When chromosomes split into two identical sets
  • Organelle: A tiny “organ” within a cell
  • Nucleus: Organelle containing genetic information, a cell’s “brain” 
  • Transcription: The act of copying of DNA to RNA 
  • Vacuole: An organelle found in all plant cells

Before we move onto another science objective, let’s look at one more concept which related to biology: animal names. This topic appears frequently on the MAT. 

The list below is organized as follows:

Animal: Male, Female, Offspring, Group, Adjective

  • Bear: Boar, sow, cub, sleuth, ursine 
  • Bee: Drone, queen, larva, swarm, apian 
  • Cat: Tom, queen, kitten, clowder, feline 
  • Cattle: Bull, cow, calf, herd, bovine 
  • Crow: Crow, crow, chick, murder, corvine
  • Deer: Buck, doe, fawn, herd, corvine 
  • Ferret: Hob, jill, kit, business, ferrety
  • Fox: Dog, vixen, kit, skulk, vulpine 
  • Goat: Billy, nanny, kid, tribe, hircine
  • Horse: Stallion, mare, foal, herd, equine 
  • Lion: Lion, lioness, cub, pride, leonine 
  • Pig: Boar, sow, piglet, sounder, porcine 
  • Rat: Buck, doe, kitten, mischief; murine 
  • Sheep: Ram, ewe, lamb, flock, ovine 
  • Snake: Snake, snake, snakelet, pit, serpentine
  • Swan: Cob, pen, cygnet, wedge, swanlike

Other interesting collective nouns include a parliament of owls, romp of otters, fluffle of rabbits, and a prickle of porcupines. Some might also call a group of snakes a nightmare, but it’s best to stick with pit on the exam!


Don’t worry, you won’t be asked to balance any chemical equations on the Miller’s Analogy Test!

However, it is a good idea to know some basic chemistry terms and to understand what they mean. Chemistry is yet another science which test-takers should not be surprised to encounter on the exam.

As a reminder, chemistry is the science of substances that make up matter. So let’s review some basic concepts.

  1. Are atoms and molecules the same thing? No. An atom is smaller than a molecule. Atoms are made up of protons, neutrons, and electrons. Molecules are a group of atoms that are bonded together?
  2. How do atoms bond together? Three types of bonds can bind atoms: ionic bonds, covalent bonds, and metallic bonds.
    • Ionic bonds: electrons transfer from one atom to another, creating oppositely charged ions that attract each other.
    • Covalent bonds: atoms share electron pairs between themselves. (Think: covalent – coworker. You share with your coworkers.)
    • Metallic bonds: if you’ve ever gathered static and been slightly shocked by touching a metal doorknob, you know that metal is a great conductor of electricity. That’s because the electrons of a metal can easily become detached or mobile. The attraction of these mobile electrons to their stationary cation forms a metallic bond.
  3. So, what’s an element? An element is a substance made entirely from the same type of atoms. All of the atoms making up an element have the same number of protons.
  4. What’s the difference between protons, neutrons, and electrons? Protons have a positive charge, neutrons have no charge, and electrons have a negative charge. (Think proton – positive, neutron – neutral, electron – being electrocuted would be a negative experience!)

Here are a handful of other helpful chemistry terms:

  • Acid: Reacts with a base, has a pH less than 7 
  • Anion: A negatively charged ion 
  • Base: Reacts with an acid, has a pH greater than 7 
  • Cation: A positively charged ion 
  • Electrolyte: Chemical compound that conducts electricity
  • pH: Measurement of acidity 
  • Solute: Put into a solvent to form a solution
  • Solvent: Dissolves a solute to form a solution
  • Valence electrons: An atom’s outermost electrons 


Geology is about more than just rocks! The field of geology covers the processes which act on the physical structure of the earth as well as the substances that make up the earth. Geologists are also concerned with the history of the earth itself.

Let’s review the earth’s time periods and eras. The numbers in the chart below reflect millions of years ago. The chart is organized from most recent to earliest.

Now, let’s look at some key terms related to geology:

  • Cementation: Process by which minerals “glue” sediments together to from rocks 
  • Clay: Mineral grains formed chemical weathering
  • Coal: Rock made of compressed and carbonized plant matter
  • Crystal: A geometric solid minerals with regular structures
  • Deposition: Process of sediment settling from water or wind
  • Faults: Cracks caused by rock movement during earthquakes
  • Lava: Molten rock that erupts from a volcano
  • Mineral: A natural chemical compound
  • Strata: Layers of rock formed by deposition
  • Tectonic plates: Portions of the lithosphere that move and affect the Earth’s crust
  • Volcanic ash: Small fragments of rock that spew from volcanoes
  • Weathering: Breakdown of surface rock; can be physical, biological, or chemical

Social Sciences

You can expect to complete to encounter the Social Sciences Content Objective as you complete analogies on the MAT. Let’s explore some branches of the social sciences which are often encountered on the exam.


Anthropology is the study of humans. There are four main subfields of anthropology: cultural anthropology, linguistic anthropology, biological anthropology, and archeology. Anthropologists explore our species as a whole, across various cultures across the world, and across multiple time periods.

Together, we’ll review, compare, and contrast some anthropological concepts.

Now that you have reviewed some terms, quiz yourself by completing this analogy:

Matriarchy : Materteral :: Patriarchy :: _______ a. Fraternal b. Sororal c. Avuncular d: Levirate

Matriarchy describes a social structure in which females are in power. Patriarchy describes a social structure in which males are in power. Materteral also relates to women; it refers to one’s aunt. So, what word refers to one’s uncle, the male equivalent of an aunt? Avuncular (C) is the correct answer.


Criminology is the study of crime. You just reviewed some anthropological terms, such as maternal, which describe the relationship to a certain family member. Did you know that there are also some terms which refer to crimes committed against certain family members?

There are terms for these crimes, especially when it comes to murder. There are other words used to describe particular murders as well. While murder is not a light topic, it can be interesting to learn about the different terms which relate to murder and other crimes.

Since criminology terms often appear in analogies classified as part of the Social Sciences Content Object, let’s review some of those words now:


Next, we’ll explore some concepts related to geography, another category that is often included on the MAT.

First, let’s look at some key geographical terms.

Now that you’ve reviewed some basic terms, let’s look at some specific locations around the world.

Are you ready to test your knowledge? Try completing the following analogy:

Longest : Missouri  :: Tallest : a. Everest b. Himalayas c. Denali d. Kilimanjaro

So, if the Missouri is the longest river in North America, you are looking for the answer choice (all of which are mountains) that describes the tallest mountain in North America. Choice C, Denali, is correct.

Political Science

It’s time to get political! Analogies related to political science are often included on the MAT as part of the Social Sciences Content Objective. 

Remember all of the -ologies that you encountered while reviewing other objectives? There are a several -acracies which are related to political science. Take a look:

  • Aristocracy: Government ruled by a few elite members (ex. patricians of Rome)
  • Autocracy: Government is ruled by one person (ex. Nazi Germany)
  • Democracy: All citizens have some influence (ex. U.S., ancient Athens)
  • Ethnocracy: Government in which one ethnic group rules over another (ex. pre-civil war southern U.S.)
  • Gerontocracy: Government ruled by elder members of population (ex. Sparta)
  • Meritocracy: Government in which those deemed to possess talent rule (ex. imperial China)
  • Mobocracy: Government ruled by the mob (ex. French Revolution)
  • Plutocracy: Government ruled by the wealthy (ex. pre-WWII Japan)
  • Ptochocracy: Government ruled by the poor (ex. Cultural Revolution in China)
  • Theocracy: Government driven by a religious institution (ex. The Vatican; fictional dystopian society in Atwood’s The Handmaid’s Tale)
  • Timocracy: Government ruled by property owners (ex. feudalism in medieval Europe)

You have already reviewed the term oligarchy. Notice that many of the terms you just read are types of oligarchies (rule by a few members). Here are a few other terms to know:

  • Capitalism: Trade and industry are controlled by private owners
  • Communism: Property is shared by the public
  • Constitutionalism: Government limited by fundamental laws
  • Socialism: Seeks to reject social classes
  • Totalitarianism: Government has complete control

So, let’s test your knowledge. Try to complete the following analogy.

Beloved : Ethnocracy :: 1984 : a. Timocracy b. Totalitarianism c. Constitutionalism d. 2019

Beloved is a novel by Toni Morrison. It is about an escaped slave in the late 18th century U.S. who flees from the South; the South was an ethnocracy during that time. George Orwell’s 1984 is a dystopian novel characterized by a totalitarian government. Choice B is the correct answer.


Now, let’s get psyched for psychology! You don’t need to prepare to become a licensed therapist in order to perform well on the MAT, but some basic psychological principles and ideas often appear on the exam.

Do you remember the guy with the big white beard? No, not Santa Claus, Sigmund Freud. Freud is often considered to be the father of psychoanalysis and a lot of modern psychology has its roots in his work.

According to Freud, everyone has an ego, an id, and a super-ego.

  • Ego: The part of an individual’s personality that is shown to the world. The ego helps an individual make sense of real life and is practical.
  • Id: The id is governed by pleasure and instinct. Freud did not believe one’s id to be freely accessible to an individual.
  • Super-ego: The super-ego is mainly unconscious and drives a person to feel guilt and to gravitate toward morality. This unconscious part of the super-ego is called the ego ideal, or the ideal self. The conscious part of the ego makes decisions.

Now that we’ve reviewed those terms, let’s talk a little bit about abnormal psychology. Abnormal psychology basically refers to concepts relating to how an individual is atypical or needs treatment for a problem. Most people find abnormal psychology to be fascinating because it describes conditions which are a bit out of the ordinary.

You have probably heard most of the following terms before, but you may be a little unsure of what they each mean. 

Let’s review:

Try out this analogy:

ADHD: Attention :: GAD : a. Depression b. Aggression c. Self-aggrandizement d. Anxiety

People suffering from ADHD struggle with attention and people with GAD struggle with anxiety; D is correct. 

Next, let’s talk about conditioning, a process during which organisms (including humans) learn. 

In classical conditioning, a stimulus is introduced and the subject learns to respond a stimulus introduced to the environment. Pavlov’s dogs are a famous example. Psychologist Ivan Pavlov performed an experiment in which whenever he fed a group of dogs, the food was accompanied by the sound of a bell.

After a time, the dogs would drool in anticipation of food whenever they heard the bell–even if no food was presented to them.

In operant conditioning, the subject has a greater role to play. The subject is responsible for changes in its environment. B.F. Skinner is the psychologist which most people associate with operant conditioning. Skinner performed an experiment in which rats who accidentally pulled a lever were rewarded with food. Soon, they were conditioned to pull the lever in order to receive food. This is an example of positive reinforcement.

Refer to the table below:


Next, let’s sharpen up your sociology skills. Sociology is the study of how people act in groups or societies. Let’s look at some key terms.

As you may notice, sociological content is very similar to anthropological content. Now that we’ve brushed up on the Social Sciences Content Objectives, let’s move onto another area.


If math isn’t your forte, you’ll probably appreciate that only about 14% of the content on the MAT is mathematical. Furthermore, you won’t have to perform any calculations on the exam. However, it is important that you understand some mathematical principles and terms related to math in order to score well on the exam.

Let’s look at some topics which are commonly included in the Mathematics Content Objectives.

Algebra and Arithmetic

The following table gives you some terms which you may encounter on the exam, as well as definitions and examples. Pay special attention to the bold text in the examples.

Try this example analogy:

Multiplication : Product :: Division :  a. Dividend b. Divisor c. Quotient d. Numerator

The product is the answer to a multiplication problem. The quotient is the answer to a division problem. C is correct.


Next, let’s review some geometric terms.

Now, let’s look at some specific types of shapes. Pay attention to their prefixes; a prefix tells you how many sides a shape has.

  • Triangle: Three sides
  • Quadrilateral: Four sides (in a square, all four sides are equal and all angles equal 90°)
  • Pentagon: Five sides
  • Hexagon: Six sides 
  • Heptagon: Seven sides
  • Octagon: Eight sides
  • Nonagon: Nine sides
  • Decagon: Ten sides
  • Hendecagon: Eleven sides
  • Dodecagon: Twelve sides

Now, try to complete this example analogy:

Triangle : Equilateral :: Quadrilateral :  a. Rectangle b. Polygon c. Square d. Scalene

An equilateral triangle is a triangle in which the sides are all equal and the angles are all equal. A square is a quadrilateral in which the sides are all equal and the angles are all equal. C is the correct answer.

Roman Numerals

Although we rarely encounter Roman numerals in our daily lives, they have a tendency to show up on the MAT. Remember: when a smaller value precedes a larger values, you should subtract the smaller value from the larger value. 

  • I: 1 
  • V: 5 
  • X: 10 
  • L: 50 
  • C: 100 
  • D: 500 
  • M: 1,000

Example: MCMLXXXIV = 1984.

And that’s a basic overview of the content objectives on the MAT!

Analogy Types

The Miller Analogies Test features analogies that fall into four major classifications:

  • Semantic 
  • Classification
  • Association
  • Non-semantic

We will look closer at these analogy types now.


Semantic analogies deal with the meanings of words. Let’s look at some types of semantic analogies.

Synonyms and Antonyms

Synonyms are two words that are basically the same in meaning, such as small and little. Practice with the following analogy:

METICULOUS : EXACTING :: CURRENCY _________________ 

  1. denomination 
  2. fiscal
  3. financial
  4. accepting

This is a synonym analogy, albeit in disguise. Meticulous and exacting can be synonyms. The word you are looking for is a synonym for currency. Choices A – C are not synonyms for currency. These choices are misleading because, like currency, they can relate to money.

However, currency can mean accepting. D is the correct answer. Be sure to consider all meanings of a word when completing an analogy, especially if none of the answers feel right at first.

Antonyms are two words that are opposite in meaning, such as wet and dry. Try the following analogy:


  1. Xenocentrism
  2. Xenophobia
  3. Assimilation
  4. Anthropocentrism

Dubious and indisputable are antonyms. Choice A, which means “preference for another culture” is an antonym for ethnocentrism. Choice A is correct.


Analogies which deal with intensity will include a term which is similar to another term, but to a greater or lesser degree. For example, hate is much stronger than dislike.

Take a look at the following analogy and pay attention to the intensity of the words:

STREAMING : TORRENTIAL :: TIRE :  _________________ 

  1. incapacitate
  2. bore
  3. doldrum
  4. tread

The correct answer is A. Picture torrential rain; it is more intense than streaming rain. To incapacitate someone can mean to completely wear them out to the point that they are not functioning; this is more intense than to tire someone.

Try another analogy:

VITRIOLIC : SATIRICAL :: EXTIRPATE :  _________________ 

  1. raze
  2. subdue
  3. mollify
  4. confound

B is the correct answer. This time, the words are ordered from mild to strong. Subdue is less intense than extirpate.


Next, let’s look at some classification analogies.


Some analogies deal with categories. You may need to classify a member of a class or to determine a subordinate or superordinate relationship by classifying the terms. 

Try this analogy:

KILIMANJARO : MOUNTAIN :: VOLGA :  _________________

  1. lake
  2. gulf
  3. river
  4. range

Kilimanjaro is classified as a mountain and the Volga is a river. Choice C is correct.

Try this analogy:

BISHOP : CARDINAL :: CORPORAL :  _________________

  1. Robin
  2. Sergeant
  3. Diocesan
  4. Private

This is an analogy that follows a superordinate : subordinate structure. In the order of Catholic officials, a bishop ranks higher than a cardinal. In the U.S. Marine Corps and the U.S. Army, a corporal ranks higher than a private. D is the correct answer.

Parts and Whole

Some analogies on the MAT are formatted as part : whole, whole : part, or part : part (wherein both parts help to make up the same whole).

See if you can complete this analogy:

SODA : BREAD :: MOZZARELLA :  _________________

  1. caprese
  2. toast
  3. Italian
  4. basil

Don’t let this analogy trick you; soda bread is a type of bread. However, soda (also called baking soda or bread soda) is also an ingredient found in bread. How do you know that the term soda is being used as an ingredient, not a category of bread? 

The answer is in the answers. Since cheese is not an answer, you can rule out that relationship.

Mozzarella (along with basil and tomatoes) is an ingredient used to make a Caprese salad. A is correct.


Next, let’s look at analogies which require you to consider some different types of associations.

Object and Characteristic

Some of the analogies on the exam will ask you to think about the characteristics of an object or person. You might need to think about an attribute an object or individual possesses or lacks or the setting in which an object or individual may be found.

Try this analogy, thinking about the characteristics associated with the terms.

HERO : SLAVE :: TIMIDITY :  _________________

  1. heroine
  2. freedom
  3. capture
  4. rescue

So, hero and slave are not really related; think about the relationship between hero and timidity instead. A hero does not have timidity (cowardice). What does a slave lack? B, freedom, is correct.


Some analogies may require you to think of the order of events or the sequence of a transformation. For example, caterpillar : butterfly shows a clear sequence, or order.

Try this analogy:

IV : V :: V :  _________________

  1. VI
  2. IV
  3. X
  4. IX

Answer A is correct. Four (IV) comes before five (V). Five comes before six (VI).

Agent and Object

Agent and object analogies require you to think about cause and effect relationships, the relationships between creators and creations, and the relationships between tools and their use. 

Practice by completing this analogy:

STICKS : DRUMMER :: BOW : ________

  1. archery
  2. cellist
  3. performance
  4. violin

This analogy follows the pattern of tool : user. Choice B is correct.

Try another agent and object analogy:


  1. gravity
  2. verticality
  3. tenacity
  4. pitch

A barometer measures pressure and a plumb measures verticality. Choice B is correct.


Non-semantic analogies do not deal with the meanings of words. So, what could they possibly address? Logic and phonetics are the areas they address the most frequently. Let’s practice.


Equality analogies typically ask you to evaluate two values which are equal or unequal. You might also need to evaluate words which are equivalent to part of other words.

Here is an example:

RETALIATE : ATE :: SPRUNG : _________

  1. spring
  2. sprint
  3. run
  4. grunt

Choice C is correct. “Ate” is part of retaliate. “Run” is part of sprung. Notice that the analogy is not semantic; the meanings of the words are unimportant.

Try this analogy next:


  1. deserter
  2. stressed
  3. asserts
  4. distressed

The word semordnilap is actually a semordnilap of palindromes; but that’s not important here. All you need to know is the pattern word : (word spelled backwards). Stressed is desserts spelled backwards! B is the correct answer.

Try one more equality analogy:

4 : 16 :: 8 : _____

  1. 12
  2. 2
  3. 64
  4. 32

Since 16 is 4 squared, 8 squared is the answer. C is correct.

Letter and Sound

Some non-semantic analogies deal with letter patterns, sound patterns, rhymes, homophones, and words with similar sounds. We’ll complete a couple of example analogies.

EIFFEL : FIZZ :: WAFFLE : ______

  1. pizzaz
  2. fluffle
  3. raffle 
  4. froze

In this analogy, you are looking for double letters. Eiffel and waffle contain double fs. Fizz and pizzaz contain double zs. A is correct. 

By the way, fluffle is a real word; you may recall from before that it refers to a group of rabbits.

Here is another practice analogy:


  1. bang
  2. frantic
  3. crush
  4. dial

This analogy deals with sounds. Specifically, it deals with alliteration, words which begin with the same consonant sounds. Crash and kerchief begin with the same sound, as do phone and frantic. B is correct.

And that’s some basic info about the analogies on the MAT!

Practice Questions & Answers

Question 1

The table below displays the percentage of permits issued by building size and geographic location in Los Angeles in 2017 and 2018. The Totals columns show the total number of permits issued for each neighborhood for the corresponding years.

For each of the following statements, select Yes if the statement is true based on the information in the table. Otherwise, select No.

Find the number of permits for each of the specified neighborhoods by multiplying the total number of buildings for those neighborhoods in 2018 by the percentages indicated in the columns which correspond to the building sizes addressed in the statement:

1.1 A

Explanation: First, add the total number of buildings for 2018 (the final column) to get 611. Next, find the total number of 5+ Family buildings for 2018 using the percentages in the appropriate column:

92.9%(27) = 25.08

25.3%(175) = 44.28

44.1%(67) = 29.55

0.3%(342) = 1.03

Add these answers to get 99.94. Next, you will need to find the percentage of 611 that 99.94 is:

99.94 = x%(611)

x = 16.36

The statement is correct because 16.36% is between 15% and 17%.

1.2 B

Explanation: Compare the number of buildings using the following formulas:

(Northern Areas 5+ Family buildings) .003 x 342 = 1.026

(Downtown LA 3-4 Family buildings) .036 x 27 = .972

1.026 > 972

1.3 B

Explanation: Find the decreases mentioned using the following formulas:

(Downtown LA 3-4 Family buildings) .091(11) – .036(27) = 1 – .97

(Central LA 5+ Family buildings) .371(140) – .253(175) = 51.94 – 44.28

You can save time by skipping the subtraction since the decrease for 5+ Family buildings in Central LA is obviously greater.

Question 2

For the past few years, the number of transactions on a site which sells shoes have been declining steadily. However, after a new marketing team took over this year, there have been promising indicators of a recovery. This year, customers responded to a survey asking them to rate, on a scale of 1 to 10 (10 being highest), their experience shopping on the site. Among these survey participants, 85% of them rated their experience 8 or higher. This is an improvement over the survey results last year, when only 30% of customers rated their satisfaction 8 or higher. The site’s marketing team is attributing this improved customer experience to the new advertising campaign. Therefore, the site is in position to reverse the recent trend and increase its number of transactions this year.

In the table below, make only two selections, one in each column.

Select Support for the statement that, if true, would provide the strongest support for the argument. Select Weaken for the statement that, if true, would most seriously weaken the argument.

2.1 D

Explanation: The satisfaction of customers who shop at the site has improved, and the marketing team believes that their new approach is responsible for this improvement. Choice D provides evidence that this is true and suggests that they may result in increased transactions. If people who do not shop at the site think better of the site now than they used to, there is a possibility that they will be converted to customers of the site.

2.2 E

Explanation: The conclusion that the number of transactions will increase rests partly on the improved customer satisfaction results. Choice B says that only a score of 10 in customer satisfaction correlates strongly with an increase in transactions. Therefore, if most of the 85% of respondents who rated their satisfaction an 8 or higher rated it 8 or 9, then the survey results, in terms of predicting increased transactions, becomes much weaker.

Question 3

The following emails are exchanged between an experienced pet fish store owner and his nephew, who has just taken over his business.

For each of the following statements, select Inferable if the statement is reasonably inferable from the information provided. Otherwise, select Not Inferable.

3.1 A

Explanation: Longhorn cowfish and cubicus boxfish are similar species, according to the brief descriptions. However, cubicus boxfish tend to be cheaper than longhorn cowfish. Because the new owner has expressed doubts that his customers will buy more expensive fish, it is reasonable to infer that he would choose cubicus boxfish over longhorn cowfish.

3.2 B

Explanation: You cannot infer this. The descriptions of the species are almost identical. There is no reason to believe the new owner would choose one over the other.

3.3 A

Explanation: You can infer that the new owner will choose coral sand over other substrates for two reasons: a) the experienced owner recommends this substrate over all others and b) the new owner mentions an interest in invertebrates, which need the calcium found in the coral sand. 

Question 4

If the new owner decides to keep offering gobies to customers, which of the following strategies would be most likely to increase the number of customers who choose this species?

  1. Increasing the price of the gobies to make them seem more exotic
  2. Adding the information that gobies are non-fussy eaters to the sign on their tank
  3. Placing the gobies in the same tank as the clownfish and prominently labeling the two species as excellent tank mates
  4. Ensuring that the part-time employees know which pH level is best for gobies
  5. Placing the gobies in a tank closer to the front of the store


Explanation: If the clownfish are popular, people will be likely to go to the clownfish tank and therefore they will see the gobies. Furthermore, people looking for fish compatible with clownfish will immediately see the gobies as an option.

Question 5

The graph shows the total payroll and number of wins for 15 popular National Football teams during the 2018 season.

Based on the graph, fill in the blank in the following statements.

5.1. The likelihood that a team with a payroll greater than $125 million won more than 10 games was _____ the likelihood that a team with a payroll less than $100 million won more than 8 games.

  1. greater than
  2. equal to
  3. less than

5.2. If a team is selected at random, the probability that it will be one of the top three teams in terms of both payroll and number of wins is _____.

  1. 2 out of 15
  2. 1 out of 5
  3. 3 out of 15
  4. 4 out of 15

5.1 A

Explanation: 3 out of the 5 teams with payrolls over $125 million won more than 10 games.
⅗ = .6

4 out of the 7 teams with payrolls less than $100 million won more than 8 games.
4/7 = .57

.6 > .57

5.2 A

Explanation: The top three teams in terms of wins have the following numbers of wins: 11, 11, 14.

However, one of the teams with 11 wins has a payroll less than both of the other teams, as well as less than two teams with less wins. Therefore, one of the teams with 11 wins is in 5th place in terms of payroll and must be eliminated. Therefore, there are 2 out of 15 teams which fit the description.

Question 6

Most of the attendees at Johnston Public Library are either children or senior citizens. The library will be offering an advanced watercolor painting class to the community. The director of the library is considering two options: offer the class on a Monday morning or offer the class on a Monday evening after public school hours, so that children will be able to attend.

However, senior citizens who visit the library are statistically more likely to indicate a preference for morning programs, so the library will be better off hosting the class in the morning.

Select only one statement for the Assumption column and only one statement for the Fact column. The Assumption column should indicate the assumption required by the argument. The Fact column should indicate a statement that, if true, would provide significant logical support for the required assumption.

6.1 D
Explanation: The argument is that the library will be better off hosting the class in the morning. Since seniors prefer morning classes, the underlying assumption that makes the most sense is that most of the people who will be going to the class are seniors.

6.2 B
Explanation: If it is a fact that most young children are not capable of using advanced watercolor techniques, the class would not be appropriate for them. Therefore, there would be no reason to host the class after school hours. This supports the argument that the library is better off hosting the class in the morning.

Question 7

The table shows food critics’ ratings of various cheeses. Outstanding cheeses are those which received a grade of 96 or higher; Excellent, the cheeses that received a grade between 91 and 95, inclusive; Very Good, the cheeses that received a grade between 86 and 90, inclusive; Good, the cheeses that received a grade between 81 and 85, inclusive; Average, the cheeses that received a grade between 71 and 80, inclusive; and Below Average, the cheeses that received a grade of 70 or lower.

N/A refers to an absence of a rating by a critic.

Select Yes if the statement is true based on the table. Otherwise, select No.

7.1 A

Explanation: A vintage is considered Average if it receives a grade between 71 and 80, inclusive. Critic 4 gave 1 Average rating and Critic 5 gave 0. Keep in mind that you would be able to sort the columns on the test.

7.2 A

Explanation: Critic 1 rated 15 cheeses, and 2 received an Outstanding rating (96 or higher). Critic 2 and Critic 5 each gave 2 cheeses an Outstanding rating, but they each rated more cheeses (20 and 23, respectively). Therefore, the probability of an Outstanding rating by Critic 1 was more likely than an Outstanding rating by any other critic.

7.3 B

Explanation: Critic 5 rated 1 French cheese a 90; the statement is incorrect.

Question 8

An animal shelter is soliciting donations from the community for its program to trap, spay/neuter, and release feral animals in order to control the populations of these animals.

The total number of the community members solicited is C, and a donation of $0.10 per month for one year from each community member solicited will fund exactly F feral animals treated for the following year.

In terms of C and F, select the expression that represents how many feral animals will be funded if half the community members solicited donate $0.15 per month each for one year, and the other half of the community members do not donate. (Column 8.1)

Next, select the expression that represents how many of the community members who were solicited must donate $0.15 per month each for one year in order to fund collectively 1.5F feral animals treated for the following year. (Column 8.2)

8.1 C

Explanation: If each of the C community members solicited donates $0.10 per month, that is, $1.20 for one year, then all the community members solicited will collectively donate $1.20C.

If half the community members solicited donate $.15 per month each for oneyear, and the other half do not give donations, that means that the total of donations for the year will be 9C.

$1.20C ÷ F feral animals = $.90C ÷ ? feral animals
? = $.90CF ÷ $1.20C
? = ¾(F)

8.2 D

Explanation: Next, let x be the number of community members who must donate $0.15 per month each for one year ($1.80 each, in total) in order to fund collectively exactly 1.5F feral animals. Set up a new proportion in order to find x:

$1.20C ÷ F feral animals = $1.80x ÷ 1.5F feral animals
2C ÷ 1 = 3x ÷ 1.5
x = C

Question 9

An entertainment analyst has predicted that the family drama film Home Again will have weak box office revenue during its initial release.

Home Again will open the same Friday as two big-budget films: the highly anticipated romance Wait for Me in Paris and Wild Goose Chase 2, the sequel to an extremely popular comedy. Furthermore, Home Again is scheduled to be shown in only half as many theaters as Wild Goose Chase 2 during the week that the films are released, so its box office potential is limited.

Select Strengthen for the statement that, if true, would most strengthen the analyst’s argument, and select Weaken for the statement that, if true, would most weaken the analyst’s argument.

9.1 A

Explanation: The analyst concludes that Home Again will not do well at the box office because of its competition and the limited number of screens on which it will be initially shown. Choice A says that family dramas, such as Home Again, do not fare well at the box office when they go up against big-budget romances, such as Wait for Me in Paris. This statement reinforces the analyst’s conclusion.

9.2 D

Explanation: Choice D weakens the argument: If Home Again will be shown on more screens after the first week, then its box office potential will increase later in its theatrical run.

Question 10

Wildlife biologist: Pumas, also known as mountain lions, face many threats. Much of their territory has been taken over by humans and they compete with other large prey animals, such as wolves and bears, for prey. Many of the deer species which pumas prey on most have experienced declines in populations which overlap with puma populations in terms of territory. This population decline has been caused by hunting.

The greatest danger for adult pumas comes from farmers, who view the cats as a threat to their cattle and other livestock; farmers have been killing pumas in great numbers over several decades. As a result, puma populations have been decreasing precipitously, and are wiped out altogether from entire regions in which they used to roam, such as most of the Appalachian mountains. Pumas will be nearly extinct in the wild within a few years.

Indicate two different statements as follows: one statement identifies an assumption required by the wildlife biologist’s argument, and the other identifies a possible fact that, if true, would provide significant logical support for the required assumption.

10.1 E

Explanation: The wildlife biologist gives many reasons why pumas are vulnerable, states that puma populations have been declining, and concludes that the species will soon be nearly extinct in the wild. The argument assumes that nothing will change in the near future in order to help puma populations thrive. Choice E identifies the assumption that conservation efforts will not alter the current trend.

10.2 C

Explanation: A fact suggesting that conservation efforts will not alter the current trend is supplied by choice C Farmers, who are the biggest threat to pumas, have resisted conservation efforts.

Question 11

What is ¼% of 54?

  1. .046
  2. .14
  3. 13.5
  4. .03


(¼%) 54 = (¼ x 54)  ÷ 100 = 13.5 ÷ 100 = .135

Round to .14.

Question 12

The total area of a warehouse is 150,000 square feet. The chart below shows warehouse units Q, R, S, and T as proportions of total warehouse area.

By approximately how many square feet does the size of Unit Q exceed that of Unit S?

  1. 12,000 square feet
  2. 13,000 square feet
  3. 13,500 square feet
  4. 14,000 square feet


Explanation: To determine the size of Unit S, first determine the size of Unit T as a percentage of the total warehouse size. Unit T occupies 14,500 square feet, or approximately 10%, of the total 150,000 square feet in the warehouse. Thus, Unit Q occupies 20% of that total (100% – 28% – 42% – 10% = 20%).

The question asks for the difference in size between Unit Q (28%) and Unit S (20%). Unit Q is 8% larger than Unit S.

150,000 total square feet x .08 = 12,000 square feet.

Question 13

Customers at Carolyn’s Cafe can select two of three appetizers—soup, chips with salsa, and garden salad— along with two of three sides—pasta salad, fruit, and bread. What is the statistical probability that any customer will select soup, garden salad, fruit, and bread?

  1. 1 in 3
  2. 1 in 6
  3. 1 in 9
  4. 1 in 12

Explanation: In each set are three member pairs. Thus the probability of selecting any pair is ⅓.

⅓ x ⅓ = 1/9

The answer would be the same for regardless of which specific menu items were ordered.

Question 14

On the xy-plane below, if the equation of w is defined as y = ½x and if point B is defined by the xy-coordinate pair (10, 0), what is the area of triangle CAB?

  1. 4 square units
  2. 10 square units
  3. 16 square units
  4. 20 square units


Explanation: The slope of w is ½ which means that every 2 units from left to right corresponds to 1 unit upward. Since angle A is a right angle, the slope of line AB must be a drop of -2 for every 1 unit from left to right. Drawing a plumb line down from point A reveals that the height of triangle CAB must be 2.

Given a base (CB) of 10 and an height of 2, the area of triangle CAB is equal to 10.

Every triangle is defined as ½ (base x height).

Question 15

If e > f, and if g > h, then

  1. e – f > g – h
  2. e – g > f – h
  3. g + h < e – f
  4. e – g < f + h
  5. f + h < e + g


Explanation: If unequal quantities (g and h) are added to unequal quantities of the same order (e and f), the result is an inequality of the same order.

Question 16

If the value of Widget Company stock drops from $50 per share to $45 per share, what is the percent of decrease?

  1. 5%
  2. 10%
  3. 12%
  4. 15%


Explanation: The amount of the decrease is $5. The percent of the decrease is 5 ÷ 50 or 10 ÷ 100, which is 10%.

Question 17

Greenfield Middle School has 1,200 students. The number of students in each of the grade levels is equivalent. The following chart reflects evaluations of the students after being assessed for meeting grade-level math expectations at the beginning of the school year.

17.1. If a Greenfield Middle School student is selected at random, there is a ______ probability that he or she exceeds grade-level expectations for math.

  1. .28
  2. .29
  3. .30
  4. .33

17.2. Based on the previous chart, _____ 8th graders need improvement.

  1. 93
  2. 124
  3. 186
  4. 372

17.1  A

Explanation: The probability is .32 for a 6th grader, .29 for a 7th grader, .24 and for an 8th grader. Since the three grades have the same number of students, the probability for a student of any grade is the average of the three probabilities.

.32 +.29 +.24 = .283

Round to .28.

17.2  B

Explanation: Since there is an equal number of students in each grade level, there are 400 students in grade 8.

400 x .31 = 124

Question 18

If ↺m↻ = m² – m, what is the value of ↺-⅔↻ + ↺⅔↻?

  1. – ²/₉
  2. – ⅔
  3. ⁴/₉
  4. ⁸⁄₉


Explanation: Substitute ⅔ and -⅔  individually for m in the defined operation.

↺⅔↻ = ⁴/₉ – ⅔ = ⁴/₉ – ⁶/₉ = – ²/₉

↺-⅔↻ = ⁴/₉ + ⅔  = ⁴/₉ + ⁶/₉ = ¹⁰⁄₉

Then, add the two results together: -²/₉ + ¹⁰⁄₉ = ⁸⁄₉.

Question 19

If u + v = w, and if u – v = z, then u =

  1. ½ (w + z)
  2. w – z
  3. w + z
  4. ½ (w – z)
  5. ½ wz


Explanation: Add the two equations to find the answer:

u + v = w
u – v = z
2u = w + z

2u = w + z simplified is u = ½ (w + z)

Question 20

In the given figure, the centers of all three circles lie on the same line. The radius of the medium-sized circle is twice that of the smallest circle. If the radius of the smallest circle is 2 inches, the boundary of the shaded area is _____ inches.

  1. 12
  2. 12π
  3. 12π


Explanation: The medium circle has a radius of 4. Therefore, the diameter of the largest circle must be 12, which makes its radius 6.

The arc of a semicircle is half the circle’s circumference or πr. The length of the boundary of the shaded region can be found by adding the arcs of the three semicircles.

2π + 4π + 6π = 12π