Plane Geometry - GRE Quantitative Reasoning
Card 0 of 1504
What is the area of a circle, one-quarter of the circumference of which is 5.5 inches?
What is the area of a circle, one-quarter of the circumference of which is 5.5 inches?
Here, you need to “solve backward” from the data you have been given. We know that 0.25C = 5.5; therefore, C = 22. In order to solve for the area, we will need the radius of the circle. This can be obtained by recalling that C = 2πr. Replacing 22 for C, we get 22 = 2πr.
Solve for r: r = 22 / 2π = 11 / π.
Now, we solve for the area: A = πr2. Replacing 11 / π for r: A = π (11 / π)2 = (121π) / (π2) = 121 / π.
Here, you need to “solve backward” from the data you have been given. We know that 0.25C = 5.5; therefore, C = 22. In order to solve for the area, we will need the radius of the circle. This can be obtained by recalling that C = 2πr. Replacing 22 for C, we get 22 = 2πr.
Solve for r: r = 22 / 2π = 11 / π.
Now, we solve for the area: A = πr2. Replacing 11 / π for r: A = π (11 / π)2 = (121π) / (π2) = 121 / π.
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Quantitative Comparison
Quantity A: Area of a right triangle with sides 7, 24, 25
Quantity B: Area of a circle with radius 5
Quantitative Comparison
Quantity A: Area of a right triangle with sides 7, 24, 25
Quantity B: Area of a circle with radius 5
Quantity A: area = base * height / 2 = 7 * 24/2 = 84
Quantity B: area = πr_2 = 25_π
Now we have to remember what π is. Using π = 3, the area is approximately 75. Using π = 3.14, the area increases a little bit, but no matter how exact an approximation for π, this area will never be larger than Quantity A.
Quantity A: area = base * height / 2 = 7 * 24/2 = 84
Quantity B: area = πr_2 = 25_π
Now we have to remember what π is. Using π = 3, the area is approximately 75. Using π = 3.14, the area increases a little bit, but no matter how exact an approximation for π, this area will never be larger than Quantity A.
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Given circle O with a diameter of 2 and square ABCD inscribed within circle O, what is the area of the shaded region?
Given circle O with a diameter of 2 and square ABCD inscribed within circle O, what is the area of the shaded region?
There are two steps to this problem: determining the area of the circle and determining the area of the square. The area of the circle is πr2 which is π(2/1)2 or π. AD is a diameter of circle O and creates two isosceles right triangles with ACD and ABD. The relationship between sides of an isosceles right triangle is 1 : 1 : √2. Thus the sides of square ABCD are √2 and the area is 2. The area of the shaded region is the area of the circle minus the area of the square, or π – 2.
There are two steps to this problem: determining the area of the circle and determining the area of the square. The area of the circle is πr2 which is π(2/1)2 or π. AD is a diameter of circle O and creates two isosceles right triangles with ACD and ABD. The relationship between sides of an isosceles right triangle is 1 : 1 : √2. Thus the sides of square ABCD are √2 and the area is 2. The area of the shaded region is the area of the circle minus the area of the square, or π – 2.
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For 
, Chelsea can get either a 
diameter pizza or two 
diameter pizzas. Which is the better deal?
For
Therefore the 16 inch pizza is the better deal.
Therefore the 16 inch pizza is the better deal.
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Circle B has a circumference of 36π. What is the area of circle A, which has a radius half the length of the radius of circle B?
Circle B has a circumference of 36π. What is the area of circle A, which has a radius half the length of the radius of circle B?
To find the radius of circle B, use the circumference formula (c = πd = 2πr):
2πr = 36π
Divide each side by 2π: r = 18
Now, if circle A has a radius half the length of that of B, A's radius is 18 / 2 = 9.
The area of a circle is πr2. Therefore, for A, it is π*92 = 81π.
To find the radius of circle B, use the circumference formula (c = πd = 2πr):
2πr = 36π
Divide each side by 2π: r = 18
Now, if circle A has a radius half the length of that of B, A's radius is 18 / 2 = 9.
The area of a circle is πr2. Therefore, for A, it is π*92 = 81π.
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In the diagram above, square ABCD is inscribed in the circle. If the area of the square is 9, what is the area of the circle?
In the diagram above, square ABCD is inscribed in the circle. If the area of the square is 9, what is the area of the circle?
If the area of the square is 9, then s2 = 9 and s = 3. If the sides thus equal 3, we can calculate the diagonals (either CB or AD) by using the 45-45-90 triangle ratio. For a side of 3, the diagonal will be 3√(2). Note that since the square is inscribed in the circle, this diagonal is also the diameter of the circle. If it is such, the radius is one half of that or 1.5√(2).
Based on that value, we can computer the circle’s area:
A = πr2 = π(1.5√(2))2 = (2.25 * 2)π = 4.5π
If the area of the square is 9, then s2 = 9 and s = 3. If the sides thus equal 3, we can calculate the diagonals (either CB or AD) by using the 45-45-90 triangle ratio. For a side of 3, the diagonal will be 3√(2). Note that since the square is inscribed in the circle, this diagonal is also the diameter of the circle. If it is such, the radius is one half of that or 1.5√(2).
Based on that value, we can computer the circle’s area:
A = πr2 = π(1.5√(2))2 = (2.25 * 2)π = 4.5π
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A small circle with radius 5 lies inside a larger circle with radius x. What is the area of the region inside the larger circle, but outside of the smaller circle, in terms of x?
A small circle with radius 5 lies inside a larger circle with radius x. What is the area of the region inside the larger circle, but outside of the smaller circle, in terms of x?
Since the answers are in terms of pi, simply find the area of each circle in terms of x and ∏:
Smaller: ∏(5)2 = 25∏
Larger: ∏x2
We must subtract the inner circle from the outer circle; this translates to ∏x2-25∏.
Since the answers are in terms of pi, simply find the area of each circle in terms of x and ∏:
Smaller: ∏(5)2 = 25∏
Larger: ∏x2
We must subtract the inner circle from the outer circle; this translates to ∏x2-25∏.
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Quantitative Comparison
Quantity A: Area of a circle with radius r
Quantity B: Perimeter of a circle with radius r
Quantitative Comparison
Quantity A: Area of a circle with radius r
Quantity B: Perimeter of a circle with radius r
Try different values for the radius to see if a pattern emerges. The formulas needed are Area = π r_2 and Perimeter = 2_πr.
If r = 1, then the Area = π and the Perimeter = 2_π_, so the perimeter is larger.
If r = 4, then the area = 16_π_ and the perimeter = 8_π_, so the area is larger.
Therefore the relationship cannot be determined from the information given.
Try different values for the radius to see if a pattern emerges. The formulas needed are Area = π r_2 and Perimeter = 2_πr.
If r = 1, then the Area = π and the Perimeter = 2_π_, so the perimeter is larger.
If r = 4, then the area = 16_π_ and the perimeter = 8_π_, so the area is larger.
Therefore the relationship cannot be determined from the information given.
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Quantitative Comparison
A circle has a radius of 2.
Quantity A: The area of the circle
Quantity B: The circumference of the circle
Quantitative Comparison
A circle has a radius of 2.
Quantity A: The area of the circle
Quantity B: The circumference of the circle
This is one of the only special cases where the area equals the circumference of the circle. The Area = πr_2 = 4_π. The circumference = 2_πr_ = 4_π_.
Note: For a quantitative comparison such as this one where the columns have numeric values instead of variables, the answer will rarely be "cannot be determined".
This is one of the only special cases where the area equals the circumference of the circle. The Area = πr_2 = 4_π. The circumference = 2_πr_ = 4_π_.
Note: For a quantitative comparison such as this one where the columns have numeric values instead of variables, the answer will rarely be "cannot be determined".
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If a circular garden with a radius of 3 ft. is bordered by a circular sidewalk that is 2 ft. wide, what is the area of the sidewalk?
If a circular garden with a radius of 3 ft. is bordered by a circular sidewalk that is 2 ft. wide, what is the area of the sidewalk?
To solve this problem, you must find the area of the entire circle (garden and sidewalk) and subtract it by the area of the inner garden. The entire area has a radius of 5 ft. (3 ft. radius of the garden plus the 2 ft. wide sidewalk), giving it an area of 
. The inner garden has a radius of 3 ft. and an area of 
. The difference is 
, which is the area of the sidewalk.
To solve this problem, you must find the area of the entire circle (garden and sidewalk) and subtract it by the area of the inner garden. The entire area has a radius of 5 ft. (3 ft. radius of the garden plus the 2 ft. wide sidewalk), giving it an area of
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If a circular monument with a radius of 30 feet is surrounded by a circular garden that is 20 feet wide, what is the area of the garden?
If a circular monument with a radius of 30 feet is surrounded by a circular garden that is 20 feet wide, what is the area of the garden?
To find the area of the garden, you need to find the entire area and subtract that by the area of the inner circle, or the monument. The radius of the larger circle is 50, which makes its area 
. The radius of the inner circle is 30, which makes its area 
. The difference is 
.
To find the area of the garden, you need to find the entire area and subtract that by the area of the inner circle, or the monument. The radius of the larger circle is 50, which makes its area
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The sum of the two bases in a parallelogram is 
An adjacent side to the bases is 
the length of one of the two base measurements. Find the length of one side that is adjacent to the bases.
The sum of the two bases in a parallelogram is
In this problem, you are told that the sum of the two bases in a parallelogram is 
Since the two bases must be equivalent, each side must equal: 
. Additionally, the problem states that the adjacent sides are 
the length of the bases. Therefore, an adjacent side to the base must equal:
In this problem, you are told that the sum of the two bases in a parallelogram is
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A parallelogram has a base of 
. The perimeter of the parallelogram is 
. Find the length of an adjacent side to the base.
A parallelogram has a base of
A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of 
and two base sides each with a length of 
Since the perimeter and one base length is provided in the question, work backwards using the perimeter formula:
, where 
and 
are the measurements of adjacent sides.
Thus, the solution is:
Check:
A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of
Thus, the solution is:
Check:
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Using the parallelogram shown above, find thesum of the two sides adjacent to the base.
Using the parallelogram shown above, find thesum of the two sides adjacent to the base.
To find one of the adjacent sides to the base, first note that the interior triangles represented by the red vertical lines must have a height of 
and a base length of 
Then, apply the formula: 
to find the length of one side.
Thus, the solution is:
Therefore, the sum of the two sides is:
To find one of the adjacent sides to the base, first note that the interior triangles represented by the red vertical lines must have a height of
Thus, the solution is:
Therefore, the sum of the two sides is:
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A parallelogram has a base of 
. The perimeter of the parallelogram is 
. Find the sum of the two adjacent sides to the base.
A parallelogram has a base of
A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of 
and two base sides each with a length of 
. In this question, you are provided with the information that the parallelogram has a base of 
and a total perimeter of 
. Thus, work backwards using the perimeter formula in order to find the sum of the two adjacent sides to the base.
, where and are the measurements of adjacent sides.
Thus, the solution is:
A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of
Thus, the solution is:
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A parallelogram has a base measurement of 
. The perimeter of the parallelogram is 
. Find the measurement of an adjacent side to the base.
A parallelogram has a base measurement of
A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of 
and two base sides each with a length of 
. In this question, you are provided with the information that the parallelogram has a base of 
and a total perimeter of 
. Thus, work backwards using the perimeter formula in order to find the length of one missing side that is adjacent to the base.
, where and are the measurements of adjacent sides.
Thus, the solution is:
A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of
Thus, the solution is:
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A parallelogram has a base of 
. The perimeter of the parallelogram is 
. Find the sum of the two adjacent sides to the base.
A parallelogram has a base of
A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of 
and two base sides each with a length of 
. In this question, you are given the information that the parallelogram has a base of 
and a total perimeter of 
. Thus, work backwards using the perimeter formula in order to find the sum of the two adjacent sides to the base.
, where and are the measurements of adjacent sides.
Thus, the solution is:
A parallelogram must have two sets of congruent/parallel opposite sides. This parallelogram must have two sides with a measurement of
Thus, the solution is:
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A parallelogram has a base of 
. An adjacent side to the base has a length of 
. Find the perimeter of the parallelogram.
A parallelogram has a base of
A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of 
and two base sides each with a length of 
. To find the perimeter of the parallelogram apply the formula:
, where and are the measurements of adjacent sides.
Thus, the solution is:
A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of
Thus, the solution is:
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A parallelogram has a base measurement of 
. The perimeter of the parallelogram is 
. Find the measurement for an adjacent side to the base.
A parallelogram has a base measurement of
A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of 
and two base sides each with a length of 
. However, to solve this problem you must first convert the provided perimeter measurement from feet to inches. Since an inch is 
of 
foot, 
feet is equal to
inches.
Now, you can work backwards using the formula:
, where and are the measurements of adjacent sides.
Thus, the solution is:
A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of
Now, you can work backwards using the formula:
Thus, the solution is:
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Using the parallelogram shown above, find the length of side 
Using the parallelogram shown above, find the length of side
A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of 
and two base sides each with a length of 
Since the perimeter and one base length is provided in the question, work backwards using the perimeter formula:
, where and are the measurements of adjacent sides.
Thus, the solution is:
A parallelogram must have two sets of congruent/parallel opposite sides. Therefore, this parallelogram must have two sides with a measurement of
Thus, the solution is:
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