GRE Area Questions Practice Questions with Answers

Concept Explanation

GRE area questions measure your ability to calculate the space enclosed within two-dimensional shapes like triangles, quadrilaterals, and circles using specific geometric formulas. These problems are a staple of the Quantitative Reasoning section, often requiring you to combine multiple formulas or derive dimensions from given perimeters or diagonal lengths. Understanding the relationship between different shapes is essential, as many high-level GRE Prep problems involve shaded regions where you must subtract the area of one shape from another.

To excel at GRE area questions, you must memorize the following fundamental formulas:

• Square: Area = $s^2$, where $s$ is the side length.
• Rectangle: Area = $l \ \times w$, where $l$ is length and $w$ is width.
• Triangle: Area = $\ \frac{1}{2} \ \times \text{base} \ \times \text{height}$.
• Circle: Area = $\pi r^2$, where $r$ is the radius.
• Parallelogram: Area = $\text{base} \ \times \text{height}$.
• Trapezoid: Area = $\ \frac{a + b}{2} \ \times h$, where $a$ and $b$ are the parallel bases.

According to Wikipedia's definition of area, the concept can be understood as the amount of material with a given thickness that would be necessary to fashion a model of the shape. On the GRE, you will frequently encounter "composite figures," which are shapes made up of two or more simpler polygons. For these, the strategy is to decompose the figure into recognizable parts, calculate their individual areas, and sum them up.

Solved Examples

1. A circle is inscribed inside a square with a side length of 10. What is the area of the region inside the square but outside the circle?
   1. Find the area of the square: $10 \ \times 10 = 100$.
   2. Determine the circle's radius: Since the circle is inscribed, its diameter equals the side of the square (10), so the radius $r = 5$.
   3. Find the area of the circle: $\pi \ \times 5^2 = 25\pi$.
   4. Subtract the circle's area from the square's area: $100 - 25\pi$.
2. A right triangle has a hypotenuse of 13 and one leg of 5. What is its area?
   1. Use the Pythagorean theorem to find the missing leg (height): $5^2 + h^2 = 13^2$.
   2. Solve for $h$: $25 + h^2 = 169 \ \rightarrow h^2 = 144 \ \rightarrow h = 12$.
   3. Apply the triangle area formula: $\ \frac{1}{2} \ \times 5 \ \times 12 = 30$.
3. The area of a trapezoid is 42. If the parallel bases are 6 and 8, what is the height?
   1. Set up the equation using the formula: $42 = \ \frac{6 + 8}{2} \ \times h$.
   2. Simplify the base calculation: $\ \frac{14}{2} = 7$.
   3. Solve for $h$: $42 = 7h \ \rightarrow h = 6$.

Practice Questions

1. A rectangle has a perimeter of 30 and a length of 9. What is the area of this rectangle?
2. If the area of a circle is $49\pi$, what is the circumference of the circle?
3. A square has a diagonal of length $6\sqrt{2}$. What is the area of the square?

4. An equilateral triangle has a side length of 8. Calculate its area.
5. A parallelogram has a base of 12 and a height of 7. A triangle with a base of 6 and a height of 7 is removed from it. What is the remaining area?
6. Quantity A: The area of a circle with radius 4. Quantity B: The area of a square with side length 7. Which is greater?
7. The length of a rectangle is increased by 20% and the width is decreased by 10%. What is the percentage change in the area?
8. A circular garden is surrounded by a path that is 2 meters wide. If the garden has a radius of 5 meters, what is the area of the path?
9. The area of a square is twice the area of a triangle with base 8 and height 9. What is the perimeter of the square?
10. A rhombus has diagonals of 10 and 24. What is the area of the rhombus?

Answers & Explanations

1. 54: Perimeter $P = 2l + 2w$. So, $30 = 2(9) + 2w$, which means $30 = 18 + 2w$, so $2w = 12$ and $w = 6$. Area $= 9 \ \times 6 = 54$.
2. $14\pi$: Area $= \pi r^2 = 49\pi$, so $r^2 = 49$ and $r = 7$. Circumference $= 2\pi r = 2\pi(7) = 14\pi$.
3. 36: In a square, the diagonal $d = s\sqrt{2}$. Thus, $s = 6$. Area $= s^2 = 6^2 = 36$.
4. $16\sqrt{3}$: The formula for the area of an equilateral triangle is $\ \frac{\sqrt{3}}{4}s^2$. Here, $\ \frac{\sqrt{3}}{4}(64) = 16\sqrt{3}$.
5. 63: Parallelogram area $= 12 \ \times 7 = 84$. Triangle area $= \ \frac{1}{2} \ \times 6 \ \times 7 = 21$. Remaining area $= 84 - 21 = 63$.
6. Quantity B: Quantity A $= \pi(4^2) = 16\pi \approx 16(3.14) = 50.24$. Quantity B $= 7^2 = 49$. Wait, let's recheck: $16 \ \times 3.14 = 50.24$. Since $50.24 > 49$, Quantity A is actually greater.
7. 8% increase: Original Area $= lw$. New Area $= (1.20l)(0.90w) = 1.08lw$. This is an 8% increase.
8. $24\pi$: The outer radius is $5 + 2 = 7$. Path area $= \pi(7^2) - \pi(5^2) = 49\pi - 25\pi = 24\pi$.
9. 24: Triangle area $= \ \frac{1}{2} \ \times 8 \ \times 9 = 36$. Square area $= 2 \ \times 36 = 72$. Side of square $s = \sqrt{72} = 6\sqrt{2}$. Perimeter $= 4 \ \times 6\sqrt{2} = 24\sqrt{2}$.
10. 120: The area of a rhombus is $\ \frac{d_1 \ \times d_2}{2}$. So, $\ \frac{10 \ \times 24}{2} = 120$.

1. If the radius of a circle is doubled, by what factor does the area increase?

• A2
• B4
• C8
• D16

Frequently Asked Questions

How do I find the area of an irregular polygon on the GRE?

Break the irregular polygon into smaller, standard shapes like rectangles and triangles. Calculate the area of each component separately and add them together to find the total area.

What is the relationship between area and perimeter?

There is no direct formula relating the two for all shapes, but for a fixed perimeter, a circle encloses the maximum area. For a fixed area, a circle has the minimum perimeter.

Do I need to memorize the value of pi for GRE area questions?

Usually, you can leave your answer in terms of $\pi$ or use the approximation 3.14. Using the AI Exam Simulator can help you practice when to estimate and when to keep the symbol.

How is the area of a shaded region calculated?

Identify the large outer shape and the inner shape(s) being removed. Calculate the area of the outer shape and subtract the areas of the inner shapes to find the remaining shaded region.

Can area be a negative number?

No, area represents a physical quantity of space and is always expressed as a non-negative value. If your calculation results in a negative number, check your subtraction order or formula application.