GRE Geometry Practice Set Practice Questions with Answers

Geometry represents approximately 15 percent of the Quantitative Reasoning section, requiring a solid grasp of shapes, angles, and coordinate systems. This GRE Geometry Practice Set provides a targeted review of the properties of lines, triangles, quadrilaterals, circles, and three-dimensional solids. By engaging with these problems, you will learn to identify the geometric patterns that the Educational Testing Service (ETS) frequently uses to test your logical reasoning and spatial visualization skills.

Concept Explanation

GRE Geometry is a sub-section of the GRE Quantitative Reasoning measure that evaluates your ability to apply the properties of points, lines, angles, and shapes to solve mathematical problems. Unlike advanced calculus or trigonometry, the GRE Prep curriculum focuses on foundational Euclidean geometry, which includes the relationships between parallel lines, the Pythagorean theorem, and the area and volume formulas for common shapes. A critical aspect of these problems is understanding that figures are not necessarily drawn to scale unless specifically stated. This means you must rely on given information—such as angle measurements or side lengths—rather than visual estimations. According to Khan Academy's geometry resources, mastering these properties is essential for high-stakes standardized testing. Key areas of focus include coordinate geometry, where you analyze shapes on a Cartesian plane using the distance and midpoint formulas, and the properties of circles, including arc length and sector area.

Solved Examples

Reviewing these solved examples will help you understand the step-by-step logic required for the GRE Geometry Practice Set.

1. Example 1: Triangles
   In a right triangle, the length of one leg is 6 and the hypotenuse is 10. What is the area of the triangle?
   
   1. Identify the missing leg using the Pythagorean theorem: $$a^2 + b^2 = c^2$$
   2. Substitute the known values: $$6^2 + b^2 = 10^2 \rightarrow 36 + b^2 = 100$$
   3. Solve for $b$: $$b^2 = 64 \rightarrow b = 8$$
   4. Calculate the area: $$\text{Area} = \frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} \times 6 \times 8 = 24$$
2. Example 2: Parallel Lines
   Line $L$ and Line $M$ are parallel. If a transversal cuts both lines and creates an interior angle of $75^{\circ}$, what is the measure of its supplementary consecutive interior angle?
   
   1. Recall that consecutive interior angles between parallel lines are supplementary, meaning they add up to $180^{\circ}$.
   2. Set up the equation: $75 + x = 180$.
   3. Solve for $x$: $x = 180 - 75 = 105^{\circ}$.
3. Example 3: Circles
   A circle has a circumference of $12\pi$. What is its area?
   
   1. Use the circumference formula to find the radius: $C = 2\pi r$.
   2. Set $12\pi = 2\pi r$, which simplifies to $r = 6$.
   3. Use the area formula: $A = \pi r^2$.
   4. Substitute the radius: $A = \pi (6)^2 = 36\pi$.

Practice Questions

1. A square has a diagonal of length $8\sqrt{2}$. What is the perimeter of the square?

2. In the coordinate plane, what is the distance between point $A(2, 3)$ and point $B(5, 7)$?

3. A rectangular box has dimensions of 4 inches, 5 inches, and 10 inches. What is the total surface area of the box in square inches?

4. The sum of the interior angles of a regular polygon is $1080^{\circ}$. How many sides does the polygon have?

5. A cylinder has a height of 8 and a base radius of 3. What is the volume of the cylinder?

6. In a triangle, the ratio of the angles is 2:3:4. What is the measure of the largest angle in degrees?

7. If the area of a circle is $49\pi$, what is the length of its diameter?

8. A line passes through the points $(0, 4)$ and $(2, 0)$. What is the slope of this line?

9. Two sides of a triangle are 7 and 10. Which of the following could be the length of the third side? (Select all that apply: 2, 4, 15, 18)

10. An isosceles triangle has two sides of length 5 and a base of length 6. What is the height of the triangle relative to the base of 6?

Answers & Explanations

1. Answer: 32. In a square, the diagonal $d$ relates to the side $s$ by the formula $d = s\sqrt{2}$. Given $8\sqrt{2} = s\sqrt{2}$, the side length $s$ is 8. The perimeter is $4s$, so $4 \times 8 = 32$.
2. Answer: 5. Use the distance formula: $d = \sqrt{(x_2 - x_1)^2 + (y_2 - y_1)^2}$. Here, $d = \sqrt{(5-2)^2 + (7-3)^2} = \sqrt{3^2 + 4^2} = \sqrt{25} = 5$.
3. Answer: 220. The surface area of a rectangular prism is $2(lw + lh + wh)$. Substituting the values: $2(4 \times 5 + 4 \times 10 + 5 \times 10) = 2(20 + 40 + 50) = 2(110) = 220$.
4. Answer: 8. The formula for the sum of interior angles is $(n-2) \times 180 = 1080$. Dividing by 180 gives $n-2 = 6$, so $n = 8$.
5. Answer: $72\pi$. The volume of a cylinder is $V = \pi r^2 h$. Substituting the values: $V = \pi (3^2) \times 8 = \pi \times 9 \times 8 = 72\pi$.
6. Answer: $80^{\circ}$. The sum of angles in a triangle is $180^{\circ}$. Let the angles be $2x, 3x,$ and $4x$. So, $2x + 3x + 4x = 180 \rightarrow 9x = 180 \rightarrow x = 20$. The largest angle is $4 \times 20 = 80^{\circ}$.
7. Answer: 14. Since $\text{Area} = \pi r^2$, we have $49\pi = \pi r^2$, so $r = 7$. The diameter is $2r$, which is 14.
8. Answer: -2. Slope $m = \frac{y_2 - y_1}{x_2 - x_1}$. Substituting the points: $m = \frac{0 - 4}{2 - 0} = \frac{-4}{2} = -2$.
9. Answer: 4 and 15. According to the Triangle Inequality Theorem, the third side $x$ must satisfy $10-7 < x < 10+7$, meaning $3 < x < 17$. Therefore, 4 and 15 are valid.
10. Answer: 4. Splitting the base of 6 in half creates two right triangles with a base of 3 and a hypotenuse of 5. Using the Pythagorean theorem: $3^2 + h^2 = 5^2 \rightarrow 9 + h^2 = 25 \rightarrow h^2 = 16 \rightarrow h = 4$.

Frequently Asked Questions

Are geometry diagrams on the GRE drawn to scale?

Most diagrams on the GRE are not drawn to scale, so you must rely on the provided text, labels, and geometric properties rather than visual measurements. Only coordinate geometry graphs are typically drawn accurately to their grid.

What are the most common triangle types on the GRE?

The GRE frequently tests right triangles, especially special right triangles like the 30-60-90 and 45-45-90 varieties. Isosceles and equilateral triangles are also common, often appearing in problems involving internal angles or symmetry.

Do I need to memorize the volume formulas for the GRE?

Yes, you should memorize the volume formulas for rectangular solids and cylinders, as they are not provided on the test. Occasionally, more complex formulas like the volume of a sphere might be provided within the question text itself.

How is coordinate geometry used in the GRE Geometry Practice Set?

Coordinate geometry combines algebra and geometry by placing shapes on an $x, y$ axis to calculate distance, slope, and intercepts. You can use the AI Question Generator to practice these specific types of coordinate-based problems.

What is the Triangle Inequality Theorem?

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side. This rule is often used in GRE Quantitative Comparison questions to determine possible side lengths.