We’re nearing the end of our introductory tour of MAT math. So far we’ve covered arithmetic, primes and divisibility (favorite topics of standardized tests everywhere), and a bit of algebra. Turning to geometry, you may be relieved to find that it’s the one math subtopic where the MAT’s structure really works in your favor. The hardest stuff you remember from high school geometry – such as proofs and constructions – has no place on the MAT, simply because it won’t fit. Instead, MAT geometry tends to be like other math areas, emphasizing recognition and recall of mathematical concepts rather than problem-solving skills.
MAT Geometry Loves Formulas
The simplest MAT geometry questions, in my opinion, are the ones that present you with a pair of formulas and a pair of terms that describe those formulas. Consider the following setup:
Rectangle : w × h :: Circle : _________
The expression w × h (width times height) is the formula for the area of a rectangle. So we need to look for an expression that gives the area of a circle. The correct formula is (D), πr2, but note that the circumference formula (2πr) was also thrown in, just as a distraction. This is standard practice on the MAT: the answer choices will include comfortingly familiar but incorrect options that give test-takers a sense of déjà vu.
Other questions will ask you about notation: the symbols conventionally used to indicate various geometric concepts.
→ : Ray :: ↔ : _________
(B) Line Segment
In geometry, there are three basic types of one-dimensional figures – shapes with length, but no width or depth. Lines have no endpoint; they extend infinitely in both directions. A ray has a single endpoint; a line segment has two. When drawing these shapes, it’s customary to use an arrow to indicate the ends that extend infinitely, but not those that are bounded by an endpoint.
With that in mind, here’s our analogy:
“A one-sided arrow is used to denote a ray; a two-sided arrow is used to denote a line.”
Consequently, our answer choice is (A). Again, typically of an MAT geometry question, the incorrect answer choices are also “geometry-flavored,” adding to the problem’s difficulty. A chord is a special type of line segment that intersects a circle at both endpoints. A tangent is a line that touches a curve (e.g., a circle) at exactly one point.
However, MAT geometry questions won’t always spell out the formula you need to apply; sometimes, you’ll need to recognize the opportunity to use a formula in the first place. In this next problem, for example, it’s evident that there’s some kind of relationship between a radius and a circumference.
3 : Radius :: _________ : Circumference
Here, the key is the formula C = 2πr (the circumference of a circle is 2π times its radius), which featured as one of our answer choices in a previous problem. Using this expression, we can figure out the analogy:
“_________ is 2π times 3, just as a circle’s circumference is 2π times its radius.”
The answer is (B), since 2π × 3 = 6π.
Here’s another, more challenging question in a similar vein:
Cylinder: 8 :: Cone : ________
(B) 2 2/3
(D) 3 3/4
We might begin a problem like this by asking if the number 8 has any special relationship to cylinders in general: “a cylinder has eight _________” or “there are eight _________ in a cylinder.” This, as you’d soon discover, is a dead end. Instead, we can try relating cylinder to cone. What properties do cylinders and cones have in common?
Well, they’re both solids, which means that each has a surface area and a volume. Of those two properties, it’s volume that holds the clue to this question. Here are the volume formulas for a cylinder and a cone of radius r and height h:
Vcylinder = πr2h
Vcone = (1/3)πr2h
Note that the only difference between the two is that factor of 1/3. A cone of a given radius and height is 1/3 the volume of the corresponding cylinder. Now let’s capture that relationship in an analogy. It might go something like this:
“_________ is 1/3 of 8, just as a cone is 1/3 of a cylinder.”
8 × 1/3 = 2 2/3, so answer (B) is our winner.
As with algebra, the geometry that appears on the MAT is largely high-school fare, meaning you’ll need to refresh your memory, but you won’t need to learn it from scratch. Properties of circles and triangles are common, and you may see a question about other simple shapes (e.g. quadrilaterals) or basic solids (e.g. cylinders, prisms, spheres). It’s possible, but unlikely, that you’ll see questions about complicated figures like nonagons or icosahedrons. If you’re trying to brush up in advance of the test, I recommend focusing on the material in our GRE Math Formulas eBook, as this corresponds to the lion’s share of MAT geometry questions.