Chapter 10

Getting into Shapes with Geometry

IN THIS CHAPTER

Bullet Getting straight with lines and angles

Bullet Working with polygons

Bullet Taking on triangles and quadrilaterals

Bullet Circumnavigating circles

Bullet 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.

Exploring Lines and Angles

The main parts of most of these shapes are lines and angles, so start with these.

GRE images are typically drawn close to scale, well enough for you to get a sense of what’s going on in the drawing. The drawing or the description will always tell you everything that you need (such as side lengths, parallel sides, right-angle boxes), so whether it’s drawn to scale really doesn’t factor in. You wouldn’t eyeball the answer anyway (except in graphs, explored in Chapter 12), so always look in the description for clues to unravel the drawing. On that note, if the drawing has a label that reads, “Figure not drawn to scale,” then it’s way off.

Lines

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 math . Perpendicular lines cross at right angles and are represented by the symbol math . A perpendicular bisector is a line that both passes through the midpoint of a line segment and is perpendicular to it.

Angles

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:

Angles can’t be negative.
Angles can’t be 0 degrees or 180 degrees.
Fractional angles, such as degrees or 179.5 degrees, are rare on the GRE. Angles are typically whole numbers, rounded to be easy to work with. If you’re plugging in a number for an angle, plug in a whole number, such as 30, 45, or 90.

Right angle

Right angles equal 90 degrees and are represented by perpendicular lines with a small box where the two lines meet.

Watch out for lines that appear to be perpendicular but really aren’t. An angle is a right angle only if the description reads, “the lines (or segments) are perpendicular” or you see the box in the angle (which is the most common). Otherwise, you can’t assume the angle is 90 degrees.

Other than the words “right angle” and “bisect,” you will probably not see the following terms, so don’t worry about memorizing words such as “obtuse” or “supplementary.” But, review the definitions so that you understand how the angles work, because that’s the key to solving almost any GRE angle problem.

Geometric representation of two angles labeled as a degrees and b degrees represented by perpendicular lines. The angles are not necessarily a right angle.

Acute angle

An acute angle is any angle greater than 0 degrees but less than 90 degrees.

Obtuse angle

An obtuse angle is any angle greater than 90 degrees but less than 180 degrees.

Complementary angles

Together, complementary angles form a right angle: 90 degrees.

Geometrical representation of complementary angle where two angles (x°, y°) add up to 90° to form a right angle.

Supplementary angles

Together, supplementary angles form a straight line: 180 degrees.

Geometrical representation of supplementary angle where two angles (x°, y°) add up to 180° to form a straight line.

Vertical angles

Vertical angles are formed when two lines cross, and they always have equal measures.

Bisectors

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 math , 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.

Geometrical representation of bisector in which a bisector named A bisects with an angle of 20 degree, a bisector named B bisects a line C and a bisector D bisects a line E perpendicularly with 90 degree.

Other key points

Angles around a point total 360 degrees, just as in a circle.

Geometrical representation that forms as a circle when connecting with four other angles.

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.

Geometrical representation that represents four equal angles when it cuts through two parallel lines.

Never assume that lines are parallel unless the question or the image states that they are. The symbol indicates that the two lines are parallel.

Recognizing Polygons

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

A polygon having all sides equal and all angles equal is called a regular polygon. For example, an equilateral triangle is a regular triangle, and a square is a regular quadrilateral. Equilateral refers to equal side lengths, as in a diamond (rhombus), and equiangular refers to equal angles, as in a rectangle. With the equilateral triangle, the equal sides make it have equal angles, but this isn’t the case with other shapes.

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.

Determining total interior angle measure

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:

Triangle:
Quadrilateral:
Pentagon:
Hexagon:
Heptagon:
Octagon:
Nonagon:
Decagon:

If you’re stuck and can’t remember the formula, just start with a triangle, which with three sides has an angle total of , and add 180 for each additional side.

Finding one interior angle

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:

Be sure that the polygon is regular or equiangular. The question will typically state this, but if it doesn’t, look for some other clue in the question to find the measure of that angle.

Understanding Triangles

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.

Recognizing triangle types

You need to know three types of triangles for the GRE: equilateral, isosceles, and right, as described in the following sections.

Equilateral triangle

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.

Isosceles triangle

An isosceles triangle has two equal sides and two equal angles.

Right triangle

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.

Noting key characteristics

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:

The largest angle is opposite the longest side. Conversely, the smallest angle is opposite the shortest side.

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The sum of any two sides must be greater than the length of the remaining side. This idea can be written as , where a, b, and c are the sides of the triangle.

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The sum of the interior angles is always 180 degrees. No matter what the triangle looks like, or how big it is, the angle total is 180 degrees.

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Any exterior angle is equal to the sum of the two other interior angles. Follow the logic. The sum of supplementary angles is , and the angle (inside the triangle, lower right) and the angle (outside the triangle) are supplementary, so . The sum of the angles inside any triangle is , so in this case . The two angles on the left ( and ) add to the angle outside the triangle, .

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Calculating perimeter and area

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.

Calculating perimeter

Perimeter is the distance around a triangle, so add up the lengths of the sides.

Geometrical representation of right triangle which has the lengths 3, 4, and 5 respectively. Perimeter is the sum of the lengths.

Calculating area

The area of a triangle is math . The height is a line perpendicular to the base, and in a right triangle it’s one of the sides.

Geometrical representation of right triangle which has 60 degree and 30 degree on the other sides with height h and breadth b.

The height may be inside the triangle, represented by a dashed line and a 90-degree box.

Geometrical representation of a triangle in which a point b is taken in the middle of the base line. An angle of 90 degrees is taken from the point b. A dashed perpendicular line is drawn from the point b to the top of the triangle and it is denoted as height h.

The height may be outside the triangle, which the GRE uses to create trick questions. Regardless, use the same formula to find the area: .

Geometrical representation of a triangle with the base b. The height h is denoted by an extended dashed line which is taken as 90 degrees to meet the top of the triangle.

Understanding the Pythagorean theorem

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.

Geometrical representation of a right triangle in which the other two angles are 60 degrees and 30 degrees. The length of the hypotenuse is 20, base is 10 square root of 3, and the height is 10.

For example, say you’re asked for the length of the base of this right triangle:

Geometrical representation of a right triangle in which the hypotenuse length is 2, the base is b and the length of the height is 1.

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:

Before calculating the value of , take a quick glance at the answer choices — they will almost always be in terms of the radical. This is an example of where the calculator is not always your friend: If you are all set with an answer of 1.732, and you have to choose from the answer choices , , and , you’ll have to start over.

Or, say you’re asked for the length of the hypotenuse of this triangle:

Geometrical representation of a right triangle in which the length of hypotenuse is indicated as c, base as b, and the length of the height as a. The values of a and b are given as 3 and 4, respectively.

Here are the calculations:

Identifying common right triangles

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.

3:4:5

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.

Geometrical representation of a right triangle in which the hypotenuse length is given as 5, the base length as 4, and the height as 3.

Because these side lengths are a ratio, the sides of a right triangle can be in any multiple of these numbers, such as (multiplied by 2), (multiplied by 3), or (multiplied by 5).

5:12:13

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 math .

Geometrical representation of a right triangle in which the hypotenuse length is given as 13, the base length as 12, and the height as 5.

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 math .

Geometrical representation of an isosceles triangle in which both the base and height are 2 and the hypotenuse length is 2 square root 2.

This formula helps when working with squares. If you know that the side of a square is 5 and you need the diagonal of the square, you know immediately that the diagonal is , because a square’s diagonal cuts the square into two isosceles right triangles This is true regardless of the size of the square: If a side length is 64, you know right away that the hypotenuse is . If the side length is , you know right away that 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:

Example If the diagonal of a square is 5, what is the area of the square?

Geometrical representation of a square in which a line is drawn diagonally from the top left corner of the square to the bottom right corner and the length of that line is given as 5.

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 math , and the side-length ratio of the triangle is this:

The hypotenuse is 5, and the length of any side of the square is math . The area of the square is then math square units.

This is a special ratio for the sides of a math triangle, which has angles measuring 30, 60, and 90 degrees. S is the length of the short side (opposite the 30-degree angle), math is the longer side, and 2s is the hypotenuse.

Geometrical representation of a right triangle in which the other two angles are 60 degree and 30 degree. The length of hypotenuse is given as 10, the base as 5, and the height as 5.

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 math 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.

Geometrical representation of two right triangles in which both triangles has the same angles of 60 degree and 30 degree. In the first triangle, the length of the hypotenuse is 20, the base is 10 and the height is 10. In the second triangle, the length of the hypotenuse is 100, base is 50, and the height is 50.

Knowing Quadrilaterals

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.

Geometrical representation of a quadrilateral which is splitted diagonally that makes two internal triangles. Each triangle has the angle of 180 degrees.

Square: The simplest quadrilateral is the square — a quadrilateral with four equal sides and four right angles. The area of a square is (that is, ), or .

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Rhombus: A rhombus is a quadrilateral with four equal sides and angles that aren’t right angles. A rhombus looks like a diamond, except that it can be sideways. The area of a rhombus is , or .

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Rectangle: A rectangle is a quadrilateral that has four right angles and opposite sides that are equal. (Rectangle means “right angle.”) The area of a rectangle is (which is the same as ).

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Parallelogram: A parallelogram is a quadrilateral where the opposite sides are equal but the angles aren’t necessarily right angles. The area of a parallelogram is , and the height is the distance between the two bases. The height is represented by a perpendicular dashed line from the tallest point of the figure down to the base.

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A square is a specific rectangle, and a rectangle is a specific parallelogram.
Trapezoid: A trapezoid is a quadrilateral that has only two parallel sides, while the other two opposite sides are not parallel. The area of a trapezoid is . It doesn’t matter which base is base₁ or base₂, because you’re adding them together anyway. Just be sure to add them before you multiply by .

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Other quadrilaterals: Some quadrilaterals don’t have nice, neat shapes or special names.

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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.

Working with Circles

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.

Recognizing a circle’s features

Circles have several unique features. You need to know these to decipher the terminology used on the test.

Center: The center is right in the middle of the circle. If a question refers to the circle by a capital letter, that’s both the circle’s center and its name.

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Diameter: The diameter is the length of a line segment that passes through the center of the circle and touches the opposite sides. The diameter is equal to twice the radius.

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Circumference: The circumference is the distance, or perimeter, around the circle.

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, or pi: Pronounced “pie,” is the ratio of the circumference of a circle to the diameter of a circle. It equals approximately 3.14, but usually circle-based questions have answer choices in terms of , such as , rather than 6.28.
Radius: The radius is the distance from the center of the circle to the edge of the circle. The radius of a circle is half the diameter.

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Tangent: A tangent is a line outside a circle that touches the circle at one point.

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Chord: A chord is a line segment that crosses through the circle and connects it on two points. The longest chord in a circle is the diameter.

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Mathematical representation of Quantity A that is area of circle of radius 6, and Quantity B is area of a circle with a longest chord measuring 12.

ovalA Quantity A is greater.

ovalB Quantity B is greater.

ovalC The two quantities are equal.

ovalD 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).

Mathematical representation of Quantity A that is area of circle of radius 10, and Quantity B is area of a circle of chord 20.

ovalA Quantity A is greater.

ovalB Quantity B is greater.

ovalC The two quantities are equal.

ovalD 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).

Mastering essential formulas

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: or

Circumference is a fancy word for perimeter — the distance around a circle. The formula is math (where C is the circumference and r is radius). Because diameter is twice the radius, you may also use the equation math (where d is the diameter).

Geometrical representation of a circle in which a line is drawn from the center of the circle to a point on the edge of the circle. The surface of the circle is indicated by circumference.

For example, the circumference of a circle with a radius of 3 is

The answer choices to circle problems are almost always in terms of , so in this example, you’d select . If you do encounter a problem with the answer choices in regular numbers, such as, “How far did a wheel with a 3-inch radius travel if it rolled 6 times,” just replace the with 3.14 in your answer. But first, look at the answer choices.

Example A wagon has a wheel radius of 6 inches. If the wagon wheel travels 100 revolutions, approximately how many feet has the wagon rolled?

ovalA 325

ovalB 314

ovalC 255

ovalD 201

ovalE 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: math . 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 math 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:

Area is the space inside the circle. The formula is math , so if a circle has a radius of 4, you can find the area like this:

Geometrical representation of a circle in which a line is drawn from the center of the circle to a point on the edge of the circle and the length of the line is given as 4.

Arcs and angles inside a circle

An arc is a part of the circumference of a circle. This is typically formed by either a central angle or an inscribed angle:

Central angle: A central angle has its vertex at the center of a circle and its endpoints on the circumference. The degree measure of a central angle is the same as the degree measure of its minor arc (the smaller arc formed by the two points where the angle’s lines touch the circle).

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Inscribed angle: An inscribed angle has both its vertex and endpoints on the circumference. The degree measure of an inscribed angle is half the degree measure of its intercepted arc, as shown in the following figure. The intercepted arc is 80 degrees, and the inscribed angle is half of that, at 40 degrees.

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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.

Example In this figure, if minor arc math , what is the sum of the degree measures of angles math ?

Geometrical representation of a circle in which a point from the edge of the circle is noted as Y and another point X is noted opposite to Y. Both X and Y are connected to 5 different points, namely, a, b, c, d and e.

ovalA 60

ovalB 90

ovalC 120

ovalD 150

ovalE 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 math . The correct answer is Choice (D).

With a central angle, the resulting arc has the same degree measure as the angle. With an inscribed angle, the resulting arc has twice the degree measure of the angle.

Geometrical representation of three circles which represents an angle from the point A to C are same for those 3 circles. In the first circle, the center point is noted as B. B and C are connected. A perpendicular line is drawn from BC and the point where it meets on the circle is noted as A. Point D is taken opposite to point C. A line is drawn from D to A which is taken 45 degree from the line DC. In the second circle, the center point is noted as B. B and C are connected. A perpendicular line is drawn from BC and the point where it meets on the circle is noted as A. In the third circle, a point D is taken opposite to the arc AC. A line is drawn with an angle of 40 degree from DC.

To find the length of an arc when you have its degree measure, follow these steps:

Find the circumference of the entire circle.
Put the degree measure of the arc over 360 and reduce the fraction.
Multiply the circumference by the fraction.

Example Find the length of minor arc AC.

Geometrical representation of a circle in which the center of the circle is noted as B and a line is drawn from B to meet at a point C on the edge of the circle. An angle of 60 degrees is taken from the line BC and the point it meets on the circle is noted as A.

ovalA math

ovalB math

ovalC math

ovalD math

ovalE math

Take the steps one at a time:

Find the circumference of the entire circle:

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:
Multiply the circumference by the fraction:

The correct answer is Choice (E).

Try another one. Make it intuitive.

Example Find the length of minor arc RS in this figure.

Geometrical representation of a circle in which two points O and S are connected. An angle of 6 degree is taken and the point it meets on the circle is noted as R. So that the angle ROS is 6 degree.

ovalA math

ovalB math

ovalC math

ovalD math

ovalE 12

Find the circumference of the entire circle:
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:
Multiply the circumference by the fraction:

The correct answer is Choice (A).

Be careful not to confuse the degree measure of the arc with the length of the arc. The length is always part of the circumference and usually has a in it. If you picked Choice (E), 12, you found the degree measure of the arc rather than its length.

Sectors

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:

Find the area of the entire circle.
Put the degree measure of the sector over 360 and reduce the fraction.
Multiply the area by the fraction.

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:

Example Find the area of minor sector ABC.

Geometrical representation of a central angle in which a circle's center point is noted as B. The point B is connected to a point C on the edge of the circle. A line is drawn perpendicular to the line BC which meets at an edge of the circle and that point is noted as A. Angle ABC area is shaded. The distance between B and C are given as 8 which means the radius r is equal to 8.

ovalA math

ovalB math

ovalC math

ovalD math

ovalE math

Use the steps listed previously:

Find the area of the entire circle.
Put the degree measure of the sector over 360 and reduce the fraction.

The sector is 90 degrees, the same as its central angle:
Multiply the area by the fraction.

The correct answer is Choice (C).

Example Find the area of minor sector XYZ in this figure.

Graphical representation of a circle in which the center point is noted as Y. The point Y is connected with a point Z on the circle. An angle of 36 degree is taken from YZ and mark the point which meets on the circle as X. A line is drawn from Y to X. So that the angle of XYZ is 36 degree and this portion is shaded.

ovalA math

ovalB math

ovalC math

ovalD math

ovalE math

Find the area of the entire circle.
Put the degree measure of the sector over 360 and reduce the fraction.

A sector has the same degree measure as its intercepted arc:
Multiply the area by the fraction.

The correct answer is Choice (B).

Tackling shaded-area problems

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.

Example A circle of radius 4 inches is centered over an 8-inch square. Find the total shaded area.

Geometrical representation of a circle which is overlapped on a square. The circle is completely overlapped inside the circle. The remaining portion of the square other than the circle is shaded. The length between the center of the circle to a point on the edge of the circle is given as 4 inches. While the length of each side of the square is given as 8 inches.

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:
Calculate the area of the inside shape.
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:

Geometrical representation of a square which is overlapped exactly half by a circle. The balance portion of the square is shaded.

Challenge 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?

ovalA math

ovalB math

ovalC math

ovalD math

ovalE math

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 math . Only half the circle overlaps the square, so the area that you subtract is half the circle: math . The square has an area of 16, so the difference between the two shapes is math : The correct answer is Choice (D).

Geometrical representation of two circles in which the inner circle is overlapped by the outer circle. The portion between the outer edge of inner circle and the inner edge of outer circle is shaded.

Challenge 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?

ovalA math

ovalB math

ovalC math

ovalD math

ovalE math

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 math , while the small circle has an area of math . Subtract them for a sidewalk area of math . The correct answer is Choice (A).

Turning 3-D Shapes

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.

Calculating volume

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.

These 3-D volume formulas have one thing in common: . Keep that in mind to help you remember the formulas for cubes, rectangular solids, and cylinders. You simply calculate the area of the base as you normally do and then multiply by the object’s height.

Volume of 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 math . Multiply that by the cylinder’s height to get its volume: math .

Geometrical representation of cylinder with radius (r) and height (h). The volume of the cylinder is represented by V = πr2h.

Example What is the volume of a right circular cylinder having a radius of 3 and a height of 5?

ovalA math

ovalB math

ovalC math

ovalD math

ovalE math

The volume of a cylinder is found with the area of its circle times its height, or math . Just plug in the radius and height, and you have the volume:

The correct answer is Choice (C).

Challenge What is the height of a right circular cylinder having a volume of math and a radius of math ?

ovalA 2

ovalB 3

ovalC 4

ovalD 5

ovalE 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).

Rectangular solid volume

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 math . Multiply that by the box’s height to find its volume: math .

Example What is the volume of a rectangular shoebox having a length of 6, a width of 3, and a depth of 4?

ovalA 72

ovalB 60

ovalC 48

ovalD 36

ovalE 24

This one is just practice. Multiply out the dimensions for the answer: math .

The correct answer is Choice (A).

Challenge 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?

ovalA 10

ovalB 12

ovalC 15

ovalD 18

ovalE 20

You know that the volume of a rectangular solid is math 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).

Cube volume

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 math .

Geometrical representation of the volume of a cube. The edge of the cube is represented by e. The volume of the cube is V = e3.

Example What is the volume of a cube having an edge length of 4?

ovalA 16

ovalB 24

ovalC 32

ovalD 48

ovalE 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).

Calculating surface area

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.

Cube surface area

A cube has six identical faces, and each face is a square. The area of a square is math . On the cube, because you call each side an edge, the area of a face is math . Because the cube has six faces, the surface area is math , or math .