Chapter 10
IN THIS CHAPTER
Getting straight with lines and angles
Working with polygons
Taking on triangles and quadrilaterals
Circumnavigating circles
Working with 3-D shapes
Geometry is all about shapes: lines, angles, triangles, rectangles, squares, circles, cubes, and more. This chapter introduces you to the many basic shapes you’re likely to encounter on the GRE along with the strategies and equations that you’ll need to answer the questions. You also get hands-on practice answering a few example questions.
The main parts of most of these shapes are lines and angles, so start with these.
You’ve probably heard the term straight line, but in geometry, that’s redundant. By definition, a line is straight. If it curves, it’s not a line. Once in a while the GRE splits lines — er, hairs — and forces you to consider whether the line goes on forever in both directions, is a line segment, or has one endpoint and an arrow at the other end, making it a ray that goes on in one direction. But most of the time, don’t worry about it: You can usually solve the problem without worrying about how far the line goes.
Parallel lines don’t cross and are represented by the symbol . Perpendicular lines cross at right angles and are represented by the symbol . A perpendicular bisector is a line that both passes through the midpoint of a line segment and is perpendicular to it.
Angles are a common part of GRE geometry problems. An angle is the space between two lines or segments that cross or share an endpoint. Fortunately, there’s not much to understanding angles when you know the different types of angles and a few key concepts.
Finding an angle is usually a matter of simple addition or subtraction. Besides the rules in the following sections, these three rules apply to the angles on the GRE:
Right angles equal 90 degrees and are represented by perpendicular lines with a small box where the two lines meet.
An acute angle is any angle greater than 0 degrees but less than 90 degrees.
An obtuse angle is any angle greater than 90 degrees but less than 180 degrees.
Together, complementary angles form a right angle: 90 degrees.
Together, supplementary angles form a straight line: 180 degrees.
Vertical angles are formed when two lines cross, and they always have equal measures.
A bisector, or line that bisects, cuts directly down the middle, and this is a term that you need to know. Yes, more vocab. If a line (or segment) bisects an angle, it divides that angle into two equal angles; if a first segment bisects a second segment, the first one cuts the second one perfectly in half. And if the first segment bisects the second segment at , then it is a perpendicular bisector as mentioned previously, and yes, the GRE will expect you to know what that is. Don’t worry though — there will almost always be a drawing.
Angles around a point total 360 degrees, just as in a circle.
A line that cuts through two parallel lines forms two sets of four equal angles. In this drawing, all the x’s are the same, and all the y’s are the same.
A polygon is any closed shape consisting of line segments, which includes everything from a triangle (three sides) to a dodecagon (a dozen sides) and beyond. The polygons you’re most likely going to encounter on the GRE are triangles and quadrilaterals (with four sides). Table 10-1 lists the names of polygons you may bump into, but don’t get caught up with the names — these problems almost always include a drawing, so you can count the sides.
TABLE 10-1 Polygons
Number of Sides |
Name |
3 |
Triangle |
4 |
Quadrilateral (including the square, rectangle, trapezoid, and parallelogram) |
5 |
Pentagon |
6 |
Hexagon (think of x in six and x in hex) |
7 |
Heptagon |
8 |
Octagon |
9 |
Nonagon |
10 |
Decagon |
If two polygons are congruent, they’re identical. If they’re similar, they have identical angles but are different sizes. The following explains what you need to know about polygons for the GRE.
Because you may be asked to find the total interior angle measure of a particular polygon, keep this formula in mind (where n stands for the number of sides):
For example, here are the sums of the interior angles of the following polygons:
I’ve seen GRE problems asking for the measure of one interior angle of a regular polygon. If you see one of these problems, here’s what you do:
Remember that n stands for the number of sides (which is the same as the number of angles), so here’s how to find a single angle measure of a regular pentagon:
Comprised of three sides, the triangle is a key figure in geometry, especially on the GRE. Understanding how triangles work helps understand other polygons. The following sections introduce you to certain common triangles and explain how to do the related math.
You need to know three types of triangles for the GRE: equilateral, isosceles, and right, as described in the following sections.
An equilateral triangle has three equal sides and three equal angles. Though technically also a regular or equiangular triangle, it’s referred to as equilateral on the GRE.
In these figures, the curved lines with the double lines through them inside the triangle indicate that the angles are equal. The short lines through the sides of the triangle indicate that the sides are equal.
An isosceles triangle has two equal sides and two equal angles.
A right triangle has one 90-degree angle.
The little box in the lower-left corner of the triangle indicates that the angle is 90 degrees. If the box isn’t shown and the question doesn’t state that you’re looking at a right triangle, then don’t assume you are.
Triangles have some notable characteristics that help you field some of the geometry questions on the exam. The following points bring you up to speed:
You may encounter at least one question on the exam that asks for the perimeter or area of a triangle. The following sections can help you clear that hurdle.
Perimeter is the distance around a triangle, so add up the lengths of the sides.
The area of a triangle is . The height is a line perpendicular to the base, and in a right triangle it’s one of the sides.
The height may be inside the triangle, represented by a dashed line and a 90-degree box.
The Pythagorean theorem only works on a right triangle. It states that the sum of the squares of the two shorter sides is equal to the square of the hypotenuse. If you know the lengths of any two sides, you can find the length of the third side with this formula:
Here, a and b are the legs of the triangle, and c is the hypotenuse. The hypotenuse is always opposite the 90-degree angle and is always the longest side of the triangle.
To find the unknown length of the third side, start with the Pythagorean Theorem:
In this case you know the a and the c, but you’re missing the b. Plug in the side lengths that you have and solve for the missing side:
Or, say you’re asked for the length of the hypotenuse of this triangle:
Here are the calculations:
The two preceding triangles are examples of common right triangles. With a common right triangle, you don’t need to work the Pythagorean theorem every time the question asks for a third side length. These next sections show you four common right triangles that make things a whole lot easier.
If one side of a right triangle is 3 and the other is 4, then the hypotenuse must be 5. Likewise, if the hypotenuse is 5 and the length of one side is 4, the other side must be 3. I proved this relationship with the Pythagorean theorem in the preceding section.
If one side of a right triangle is 5 and another side is 12, then the hypotenuse must be 13. Likewise, if you know the hypotenuse is 13 and one of the sides is 5, then the other side must be 12. You don’t need the Pythagorean theorem. This triangle is not nearly as common as the 3:4:5 described in the preceding section, and it almost never appears in multiples such as .
This ratio is for an isosceles right triangle (containing 45-45-90-degree angles), and s stands for side. If one side is 2 and a second side is also 2, then the hypotenuse is .
If you’re given the length of the hypotenuse of an isosceles right triangle and need to find the length of the other two sides, use this formula, where h is the hypotenuse:
If the diagonal of a square is 5, what is the area of the square?
To find the area, first you need the length of a side. If the diagonal is 5, then right away you know the side length is , and the side-length ratio of the triangle is this:
The hypotenuse is 5, and the length of any side of the square is . The area of the square is then square units.
This is a special ratio for the sides of a triangle, which has angles measuring 30, 60, and 90 degrees. S is the length of the short side (opposite the 30-degree angle), is the longer side, and 2s is the hypotenuse.
This type of triangle is a favorite of the test-makers. The important fact to keep in mind here is that the hypotenuse is twice the length of the short side (opposite the 30-degree angle). If you encounter a word problem that says, “Given a triangle with a hypotenuse of 20, find the area,” or “Given a 30:60:90 triangle with a hypotenuse of 100, find the perimeter,” you can do so because you can quickly find the lengths of the other sides.
Any four-sided shape is a quadrilateral. The interior angles of any quadrilateral total 360 degrees, and you can cut any quadrilateral into two triangles, each of which has total interior angles of 180 degrees. Specific quadrilaterals include squares, parallelograms, trapezoids, and other shapes. The following sections present everything you need to know about quadrilaterals for the GRE.
A square is a specific rectangle, and a rectangle is a specific parallelogram.
Don’t immediately decide that you can’t find the area of a strange shape. You may be able to divide the quadrilateral into two triangles and find the area of each triangle.
Determining a circle’s circumference (distance around the circle) or area using its radius (the distance from the center of the circle to its edge) is easy if you know the formulas and the features of the circle. The following sections cover the characteristics and the formulas you need to solve problems related to circles.
Circles have several unique features. You need to know these to decipher the terminology used on the test.
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
The longest chord of a circle is the diameter. Because the diameter is twice the radius, a circle with diameter 12 has a radius of 6. In other words, the circles are the same size and have the same area. (Don’t worry about solving for that area; you know the two circles are the same.) The correct answer is Choice (C).
Quantity A is greater.
Quantity B is greater.
The two quantities are equal.
The relationship cannot be determined from the information given.
A chord connects any two points on a circle, but you don’t know whether this particular chord is the diameter. Quantity B doesn’t specify the longest chord. If it did, the quantities would be equal, but it doesn’t, so you don’t know the size of the circle in Quantity B. The correct answer is Choice (D).
Solving circle problems is all about knowing formulas and how to use them. The following sections reveal the formulas and provide guidance on solving problems, complete with sample questions.
Circumference is a fancy word for perimeter — the distance around a circle. The formula is (where C is the circumference and r is radius). Because diameter is twice the radius, you may also use the equation (where d is the diameter).
For example, the circumference of a circle with a radius of 3 is
A wagon has a wheel radius of 6 inches. If the wagon wheel travels 100 revolutions, approximately how many feet has the wagon rolled?
325
314
255
201
It cannot be determined from the information given.
The question gives the radius in inches, but the answer choices are all in feet, so the first order of business is to convert inches into feet: . One revolution of the wheel carries the wagon a distance of one circumference of the wheel, so start by finding the circumference of the circle in feet:
Then multiply by the number of revolutions:
Now look at the answer choices. They’re in real numbers, so replace with 3.14 and multiply:
Choice (E) is definitely not the answer. If you have a radius, you can solve for nearly anything having to do with circles. The correct answer is Choice (B).
Area is the space inside the circle. The formula is , so if a circle has a radius of 4, you can find the area like this:
An arc is a part of the circumference of a circle. This is typically formed by either a central angle or an inscribed angle:
Even a drawing that looks like a dream-catcher, with lines all over, isn’t so bad if you know how the vertices, arcs, and angles work.
In this figure, if minor arc , what is the sum of the degree measures of angles ?
60
90
120
150
180
The angles a, b, c, d, and e are inscribed angles, meaning each angle has half the degree measure of its intercepted arc. You know from the drawing that the endpoints of each angle are X and Y, and the problem tells you that arc XY measures 60 degrees. This means that each angle is 30 degrees, for a total of . The correct answer is Choice (D).
To find the length of an arc when you have its degree measure, follow these steps:
Find the length of minor arc AC.
Take the steps one at a time:
Don’t multiply out; the answer choices are in terms of .
Put the degree measure of the arc over 360 and reduce the fraction.
The degree measure of the arc is the same as its central angle, 60 degrees:
The correct answer is Choice (E).
Try another one. Make it intuitive.
Find the length of minor arc RS in this figure.
12
Put the degree measure of the arc over 360 and reduce the fraction.
Here, the inscribed angle is 6 degrees. Because the intercepted arc has twice the degree measure of the inscribed angle, the arc is 12 degrees:
The correct answer is Choice (A).
A sector is part of the area of a circle that comes from a central angle. The degree measure of a sector is the same as the degree measure of the central angle. You won’t see a sector from an inscribed angle.
To find the area of a sector, do the following:
Finding the area of a sector is similar to finding the length of an arc. The only difference is in the first step: You find the circle’s area, not circumference. With this in mind, try out a few sample sector problems:
Find the area of minor sector ABC.
Use the steps listed previously:
Put the degree measure of the sector over 360 and reduce the fraction.
The sector is 90 degrees, the same as its central angle:
The correct answer is Choice (C).
Find the area of minor sector XYZ in this figure.
Put the degree measure of the sector over 360 and reduce the fraction.
A sector has the same degree measure as its intercepted arc:
The correct answer is Choice (B).
A shaded-area problem presents two shapes with one overlapping the other but not completely covering it. The visible part of the shape underneath is the shaded area. Your job is to calculate the total shaded area. Fortunately, you don’t always have to find an exact number.
A circle of radius 4 inches is centered over an 8-inch square. Find the total shaded area.
This is just a basic shape that’s partly covered up. Here’s how to calculate the area of the shaded part:
Calculate the total area of the outside shape.
Each side of the square is 8 inches:
Subtract the area of the inside shape from the area of the outside shape.
The difference between the two shapes is the shaded area.
And that’s the answer!
You don’t have to get an exact number. Try one like it:
In the preceding drawing, exactly half the circle overlaps the square. If the square has a side length of 4 and the circle has a radius of 2, what is the area of the shaded region?
The key here is that half the circle overlaps the square, so you need to calculate the area of the sector to subtract from the area of the square. The circle has a radius of 2, so its area is . Only half the circle overlaps the square, so the area that you subtract is half the circle: . The square has an area of 16, so the difference between the two shapes is : The correct answer is Choice (D).
The preceding drawing shows a circular rock garden surrounded by a sidewalk. If the rock garden has a diameter of 8 and the sidewalk is exactly 2 feet wide, what is the area of the sidewalk?
This one is exactly the same as the others, only you’re subtracting two circles. The small circle has a diameter of 8, so its radius is 4. The visible part of the large circle has a width of 2, so combined with the small circle’s radius of 4, it has a total radius of 6. Now find the areas so you can subtract them. The large circle has an area of , while the small circle has an area of . Subtract them for a sidewalk area of . The correct answer is Choice (A).
There are only two 3-D shapes that you may encounter on the GRE: the rectangular solid (which includes the cube) and the cylinder. In the following sections, I provide all you need to answer any GRE question based on these shapes.
Volume is the space inside a three-dimensional object. When the GRE asks you the volume of a rectangular solid, think of how much water goes into a fish aquarium (a rectangular aquarium, not a round or curved one). The following sections explain how to calculate the volumes of the 3-D shapes you encounter on the GRE — a cube, a rectangular solid, and a cylinder.
The GRE calls its cylinders right circular cylinders, meaning that the top and bottom circles are the same and that the sides go straight up and down. A cylinder is basically a can of soup. The base of the cylinder, as a circle, has an area of . Multiply that by the cylinder’s height to get its volume: .
What is the volume of a right circular cylinder having a radius of 3 and a height of 5?
The volume of a cylinder is found with the area of its circle times its height, or . Just plug in the radius and height, and you have the volume:
The correct answer is Choice (C).
What is the height of a right circular cylinder having a volume of and a radius of ?
2
3
4
5
6
If this question seems to laugh at you, you can laugh right back. Just place the volume and radius into the formula and solve for the height:
The correct answer is choice (C).
A rectangular solid on the GRE has six sides that are rectangles. It’s basically a shoebox. The base is a rectangle, which has an area of . Multiply that by the box’s height to find its volume: .
What is the volume of a rectangular shoebox having a length of 6, a width of 3, and a depth of 4?
72
60
48
36
24
This one is just practice. Multiply out the dimensions for the answer: .
The correct answer is Choice (A).
If a rectangular salt-water aquarium has a volume of 14,400 cubic inches, and the surface area of each end is 120 square inches, what is its length, in feet?
10
12
15
18
20
You know that the volume of a rectangular solid is and in this case is 14,400. You don’t actually get the width and height, but you get their product (result from multiplying), which is 120. That’s okay, because you multiply them anyway in the formula. First find the length in inches, then convert it to feet:
Good thing you get an on-screen calculator. Anyway, if the length is 120 inches, then it’s 10 feet, because there are 12 inches in one foot. The correct answer is Choice (A).
A cube is a three-dimensional square. Think of a six-sided die (one of a pair of dice). The cube’s dimensions, length, width, and height, are the same, so they’re called edges and are all represented by the letter e. The volume of a cube is thus .
What is the volume of a cube having an edge length of 4?
16
24
32
48
64
Consider this question to be your hors d’oeuvre, or appetizer, for the true cube questions coming up. You have the edge length, so plug that into the formula:
The correct answer is Choice (E).
Surface area is the outside of an object. If you were to wrap the object as a gift, it would be how much wrapping paper you’d need. The cube is typically the only shape for which the GRE asks you to calculate the surface area. In fact, it’ll give you the cube’s volume, and you back-solve to find the surface area — or vice versa. This next section provides the formula for calculating the surface area of the cube and back-solving it to get the edge.
A cube has six identical faces, and each face is a square. The area of a square is . On the cube, because you call each side an edge, the area of a face is . Because the cube has six faces, the surface area is , or .
What is the surface area of a cube having an edge length of 3?
18
27
36
45
54
Another hors d’oeuvre. You have the edge length, so plug that in: =
The correct answer is Choice (E).
If a cube has a volume of 8, what is its surface area?
8
12
16
24
32
Now, this is more like it. First back-solve with the volume formula to find the edge length:
Now plug that into the surface area formula:
The correct answer is Choice (D).
If a cube has a surface area of 600, what is its volume?
10
60
100
600
1,000
First, back-solve with the surface area formula to find the edge length:
Now plug that puppy right back into the volume formula:
The correct answer is Choice (E).