Hard GRE Area Questions Practice Questions

A square with a side length of 4 units contains a smaller circle tangent to all four sides, and calculating the area of the remaining corners requires subtracting the circle's area from the square's total space. Hard GRE area questions often present such composite figures or shaded regions that demand a multi-step geometric approach. These problems test your ability to decompose complex shapes into basic polygons and circles, applying properties of Euclidean geometry to find dimensions that aren't explicitly stated. By mastering these high-level concepts, you can navigate the quantitative section with greater precision.

Concept Explanation

Hard GRE area questions focus on the measurement of two-dimensional space within complex, non-standard, or overlapping geometric figures. To solve these, you must be proficient with the area formulas for circles $A = \pi r^2$, triangles $A = \frac{1}{2}bh$, and quadrilaterals like trapezoids $A = \frac{1}{2}(b_1 + b_2)h$. Beyond simple formulas, these problems often involve the Pythagorean theorem to find missing side lengths or the properties of special right triangles (30-60-90 and 45-45-90). You will frequently encounter "shaded region" problems where the strategy is to calculate the area of a larger shape and subtract the area of an inner shape. Additionally, understanding the relationship between the ratio of side lengths and the ratio of areas—where the area ratio is the square of the side ratio—is essential for solving similarity-based problems in GRE Prep.

Solved Examples

1. Example 1: The Inscribed Circle
   A square is inscribed in a circle with an area of $16\pi$. What is the area of the square?
   1. Find the radius of the circle using the area formula: $16\pi = \pi r^2$, so $r^2 = 16$ and $r = 4$.
   2. The diameter of the circle is $2r = 8$. In an inscribed square, the diameter of the circle is equal to the diagonal of the square.
   3. Let the side of the square be $s$. Using the Pythagorean theorem for the diagonal: $s^2 + s^2 = 8^2$.
   4. Simplify: $2s^2 = 64$, which means $s^2 = 32$.
   5. The area of the square is $s^2$, so the area is 32.
2. Example 2: Shaded Region in a Trapezoid
   In a trapezoid $ABCD$, $AB$ is parallel to $CD$. $AB = 4$, $CD = 10$, and the height is 6. A semicircle with a diameter of 4 is placed inside the trapezoid with its diameter resting on the base $CD$. What is the area of the trapezoid not covered by the semicircle?
   1. Calculate the area of the trapezoid: $A = \frac{1}{2}(4 + 10) \times 6 = \frac{1}{2}(14) \times 6 = 42$.
   2. Find the area of the semicircle. The diameter is 4, so the radius $r = 2$.
   3. Area of a full circle is $\pi(2)^2 = 4\pi$. The semicircle is $\frac{4\pi}{2} = 2\pi$.
   4. Subtract the semicircle from the trapezoid: $42 - 2\pi$.
3. Example 3: Ratio of Areas
   Triangle $ABC$ is similar to triangle $DEF$. If the ratio of the length of $AB$ to $DE$ is $3:5$, and the area of triangle $ABC$ is 18, what is the area of triangle $DEF$?
   1. Identify the scale factor between the sides: $k = \frac{5}{3}$.
   2. The ratio of the areas is the square of the scale factor: $k^2 = (\frac{5}{3})^2 = \frac{25}{9}$.
   3. Set up the proportion: $\frac{ \text{Area of } DEF}{18} = \frac{25}{9}$.
   4. Solve for the area: $\text{Area of } DEF = 18 \times \frac{25}{9} = 2 \times 25 = 50$.

Practice Questions

1. A circle is inscribed in an equilateral triangle with a side length of 12. What is the area of the circle?

2. A rectangular path of uniform width 2 meters surrounds a rectangular garden that measures 10 meters by 15 meters. What is the area of the path?

3. In a coordinate plane, a triangle has vertices at $(0,0)$, $(8,0)$, and $(3,6)$. What is the area of this triangle?

4. A sector of a circle has an arc length of $4\pi$ and a radius of 6. What is the area of the sector?

5. A right triangle has a hypotenuse of 13 and one leg of 5. A square is built using the other leg as its diagonal. What is the area of this square?

6. Two concentric circles have radii of 5 and 10. What is the area of the ring (annulus) formed between them?

7. A rhombus has diagonals of length 12 and 16. What is its area?

8. A square and a circle have the same perimeter. Which has the larger area, and what is the ratio of the area of the circle to the area of the square?

9. A regular hexagon is inscribed in a circle with a radius of 4. What is the area of the hexagon?

10. If the area of a circle is tripled, by what factor is the radius increased?

Answers & Explanations

1. Answer: $12\pi$
   In an equilateral triangle, the height $h = \frac{s\sqrt{3}}{2} = \frac{12\sqrt{3}}{2} = 6\sqrt{3}$. The radius of the inscribed circle (incenter) is $\frac{1}{3}$ of the height: $r = 2\sqrt{3}$. Area $= \pi(2\sqrt{3})^2 = 12\pi$.
2. Answer: 116
   The outer rectangle dimensions are $(10 + 2 + 2)$ and $(15 + 2 + 2)$, which is $14 \times 19 = 266$. The inner garden is $10 \times 15 = 150$. Path area $= 266 - 150 = 116$.
3. Answer: 24
   The base of the triangle is the distance from $(0,0)$ to $(8,0)$, which is 8. The height is the y-coordinate of the third vertex, which is 6. Area $= \frac{1}{2} \times 8 \times 6 = 24$.
4. Answer: $12\pi$
   Area of a sector can be found by $A = \frac{1}{2} r L$, where $L$ is arc length. $A = \frac{1}{2} \times 6 \times 4\pi = 12\pi$.
5. Answer: 72
   Using Pythagorean theorem, the missing leg is $\sqrt{13^2 - 5^2} = \sqrt{144} = 12$. If the diagonal of a square is 12, then $s\sqrt{2} = 12$, so $s^2 = \frac{12^2}{2} = \frac{144}{2} = 72$.
6. Answer: $75\pi$
   Area of outer circle $= 100\pi$. Area of inner circle $= 25\pi$. Difference $= 75\pi$.
7. Answer: 96
   The area of a rhombus is $\frac{1}{2} d_1 d_2$. So, $\frac{1}{2} \times 12 \times 16 = 96$.
8. Answer: Circle; $\frac{4}{\pi}$
   Let perimeter $P = 4s = 2\pi r$. Then $s = \frac{\pi r}{2}$. Area of square $= s^2 = \frac{\pi^2 r^2}{4}$. Area of circle $= \pi r^2$. Ratio $\frac{\pi r^2}{\frac{\pi^2 r^2}{4}} = \frac{4}{\pi} \approx 1.27$.
9. Answer: $24\sqrt{3}$
   A regular hexagon consists of 6 equilateral triangles with side length equal to the circle's radius (4). Area of one triangle $= \frac{4^2\sqrt{3}}{4} = 4\sqrt{3}$. Total area $= 6 \times 4\sqrt{3} = 24\sqrt{3}$.
10. Answer: $\sqrt{3}$
    New area $A' = 3A$. Since $A \propto r^2$, $3\pi r^2 = \pi (r')^2$. Thus $(r')^2 = 3r^2$, and $r' = r\sqrt{3}$.

1. If the length of a rectangle is increased by 20% and the width is decreased by 20%, what happens to the area?

• AIt stays the same
• BIt increases by 4%
• CIt decreases by 4%
• DIt decreases by 2%

Frequently Asked Questions

How do I find the area of a shaded region on the GRE?

To find the area of a shaded region, identify the larger shape and the smaller shapes contained within it. Calculate the area of the larger shape and subtract the areas of the unshaded shapes to find the remaining difference.

What is the relationship between side length ratios and area ratios?

For any two similar figures, the ratio of their areas is equal to the square of the ratio of their corresponding side lengths. For example, if sides are in a $2:3$ ratio, the areas are in a $4:9$ ratio.

Does the GRE provide geometric formulas during the test?

The GRE does not provide a formula sheet, so you must memorize basic area formulas for circles, triangles, and common quadrilaterals. You can practice these with GRE practice questions with explanations to ensure recall.

How do I calculate the area of a trapezoid if the height isn't given?

If the height is missing, you can often find it by dropping perpendicular lines from the top vertices to the base, creating right triangles. Use the Pythagorean theorem or trigonometric properties to solve for the vertical height.

What is the most common mistake in area questions?

The most common error is forgetting to square the radius in circle formulas or forgetting the $\frac{1}{2}$ factor in triangle and trapezoid formulas. Using an AI Exam Simulator can help you identify and correct these calculation habits.