Calculating the area of a circle - GMAT Quantitative Reasoning

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Question

Consider the Circle :

(Figure not drawn to scale.)

If Circle represents the bottom of a silo, what is the area of the base of the silo in square meters?

Answer

The area of circle is found by the following equation:

$A=\pi r^2$

where is our radius, which is .

So,

$A=\pi*(15:m)^2=225 \pi:m^2$

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Question

A circle on the coordinate plane is defined by the equation $x^{2}+y^{2}=25$ . What is the area of the circle?

Answer

The equation of a circle centered at the origin of the coordinate plane is $x^{2}+y^{2}=r^{2}$ , where is the radius of the circle.

The area of the circle, in turn, is defined by the equation $A=\pi r^{2}$ .

Since we are provided with the equation $x^{2}+y^{2}=25$ , we can deduce that $r^{2}=25$ and that $A=25\pi$ .

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Question

What is the area of a circle with a diameter of ?

Answer

The area of a circle is defined by $A=\pi r^{2}$ , where is the radius of the circle. We are provided with the diameter of the circle, which is twice the length of .

If , then $r=\frac{d}{2}=\frac{20cm}{2}=10cm$

Therefore:

$A=\pi r^{2}$

$A=\pi (10cm)^{2}$

$A=100\pi cm^2$

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Question

For $10, Brandon can order either a 12"-diameter pizza, two 6"-diameter pizzas, or three 4"-diameter pizzas. Which option is the best value, assuming all pizzas are the same thickness?

Answer

For a circle, $Area = \pi r ^{2}$ .

Therefore, the area of the 12" pizza $=6^{2}\pi =36\pi$ .

The area of the two 6" pizzas $=2\times 3^{2}\pi=2\times 9\pi =18\pi$ .

The area of the three 4" pizzas = $3\times 2^{2}\pi=3\times 4\pi =12\pi$ .

The 12" pizza is the best option.

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Question

A circle is inscribed in a square with area 100. What is the area of the circle?

Answer

A square with area 100 would have a side length of 10, which is the diameter of the circle. The area of a circle is $\pi r^{2}$ , so the answer is $\pi * 5^{2} = 25\pi$ .

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Question

A circle on the coordinate plane has equation

$x^{2} + 4x + y ^{2} + 6y = 12$

What is its area?

Answer

The area of a circle is equal to $A = \pi r ^{2}$ , where is the radius.

The standard form of the area of a circle with radius and center is

$(x-h)^{2} + (y-k)^{2} = r^{2}$

Once we get the equation in standard form, we know $r ^{2}$ , which can be multiplied by $\pi$ .

Complete the squares:

$\left (\frac{4}{2} \right )^{2} = 4 ;\left (\frac{6}{2} \right )^{2} = 9$

so $x^{2} + 4x + y ^{2} + 6y = 12$ can be rewritten as follows:

$x^{2} + 4x + 4 + y ^{2} + 6y + 9 = 12 + 4 + 9$

$\left (x + 2 \right ) ^{2} +\left ( y + 3 \right ) ^{2} = 25$

Therefore, $r^{2} = 25$ and $A = \pi r ^{2} = 25 \pi$ .

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Question

Tom has a rope that is 60 feet long. Which of the following is closest to the largest area that Tom could enclose with this rope?

Answer

The largest square you could make would be ${15}'\times {15}'$ with an area of . However, the largest region that can be enclosed will be accomplished with a circle (so you don't lose distance creating the angles). This circle will have a circumference of 60 ft. This gives a radius of

$\frac{60}{2\pi} = \frac{30}{\pi}$

Then the area will be

$\pi \cdot r^2 = \pi \cdot (\frac{30}{\pi})^2 = \frac{900}{\pi} \approx 286$

This is closer to 280 than to 300

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Question

The above figure shows a square inscribed inside a circle. What is the ratio of the area of the circle to that of the square?

Answer

Let be the radius of the circle. Its area is $A _{\bigcirc } = \pi r^{2}$

The diagonal of the square is equal to the diameter of the circle, or . The area of the square is half the product of its (congruent) diagonals: $A_{\square } = \frac{1}{2} \cdot2r \cdot 2r = 2r^{2}$

This makes the ratio of the area of the circle to that of the square $\frac{A _{\bigcirc }}{A_{\square }} = \frac{\pi r^{2} }{2r^{2} } = \frac{\pi }{2 }$ .

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Question

A square has the same area as a circle with a radius of 12 inches. What is the sidelength of that square, in terms of $\pi$ ?

Answer

The area of a circle with radius 12 is

$A = \pi r^{2} = \pi \cdot 12^{2} = 144\pi$ .

This is also the area of the square, so the sidelength of that square is the square root of the area:

$s = \sqrt{A} = \sqrt{144\pi} = \sqrt{144} \cdot \sqrt{\pi}= 12\sqrt{\pi}$

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Question

What is the area of a circle which goes through the points ?

Answer

As can be seen in this diagram, the three points form a right triangle with legs of length 5 and 12.

A circle through these three points circumscribes this right triangle.

An inscribed right, or $90^{\circ }$ , angle intercepts a $2 \times 90 = 180 ^{\circ }$ arc, or a semicircle, making the hypotenuse a diameter of the circle. The diameter of the circle is therefore the hypotenuse of the right triangle, which we can find via the Pythagorean Theorem:

$d = \sqrt{ 5^{2} + 12 ^{2}} = \sqrt{ 25+ 144} = \sqrt{169} = 13$

The radius of the circle is half this, or $\frac{13}{2}$ .

The area of the circle is therefore:

$A = \pi r^{2} = \pi \cdot \left ( \frac{13}{2} \right) ^{2} =\frac{169\pi }{4}$

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Question

A circle on the coordinate plane has equation $x^{2} + y ^{2} = 49$ .

What is its area?

Answer

The equation of a circle centered at the origin is

$x^{2} + y ^{2} = r ^{2}$ ,

where is the radius of the circle.

The area of a circle is $A = \pi r ^{2}$ .

From the equation given in the question stem, we know that $r ^{2} = 49$ , so we can plug this into the area formula:

$A = \pi \cdot 49 = 49 \pi$

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Question

Refer to the above figure, in which the dot indicates the center of the larger circle. The larger circle is one yard in diameter. Give the area of the gray region in square inches.

Answer

The larger circle has diameter one yard, or 36 inches; its radius is half that, or 18 inches, so its area is

$A_{1} = \pi r ^{2} = \pi \cdot 18 ^{2} = 324 \pi$ square inches.

The diameter of the smaller circle is equal to the radius of the larger, or 18 inches; its radius is half that, or 9 inches, so its area is:

$A_{2} = \pi r ^{2} = \pi \cdot 9 ^{2} = 81 \pi$ square inches.

The area of the gray region is the difference of the two:

$A= 324 \pi - 81 \pi = 243 \pi$ square inches.

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Question

Refer to the above circle. The smaller circle has diameter feet. In terms of , give the area of the gray region in square inches.

Answer

Multiply the diameter of the smaller circle by 12 to convert it to inches - this will be , which is also the radius of the larger circle. The radius of the smaller circle will be half this, or .

Using the area formula for a circle $A = \pi r^{2}$ , we can substitute these quantities for and subtract the area of the smaller circle from that of the larger:

$A_{1} = \pi r^{2} = \pi \left (12k \right ) ^{2} = 144 \pi k^{2}$ square inches

$A_{2} = \pi r^{2} = \pi \left (6k \right ) ^{2} = 36 \pi k^{2}$ square inches

$A = A_{1} - A_{2} = 144 \pi k ^{2}- 36 \pi k ^{2} = 108 \pi k ^{2}$ square inches

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Question

A circle on the coordinate plane has equation

$x^{2} + y ^{2} = 50$

Which of the following represents its area?

Answer

The equation of a circle centered at the origin is

$x^{2} + y ^{2} = r ^{2}$

where is the radius of the circle.

The area of a circle is $A = \pi r ^{2}$ ; since in the equation of the circle, $r ^{2} = 50$ , we can substitute to get the area $A = \pi \cdot 50 = 50 \pi$ .

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Question

For Data Sufficiency questions, select either A, B, C, D, E based on the following rubric:

A) statement 1 alone is sufficient, but statement 2 alone is not sufficient to answer the question

(B) statement 2 alone is sufficient, but statement 1 alone is not sufficient to answer the question

(C) both statements taken together are sufficient to answer the question, but neither statement alone is sufficient

(D) each statement alone is sufficient

(E) statements 1 and 2 together are not sufficient, and additional data is needed to answer the question

Is the area of circle X less than the area of square Y?

1. The diameter of X is 2 less than the square of Y's diagonal.

2. The perimeter of Y is less than the circumference of X

Answer

In order to compare the areas of the square and circle, we need to find an equivalence between the radius of the circle and the side-length of the square.

Let's take the first statement. Tranlating it into math, we have:

$2r+2=(d\sqrt{2})^2$

Where r= the radius of the circle, and d= the length of the side of the square. The important point is that we DO NOT HAVE TO SOLVE THIS. All we need to recognize is that the equation makes a comparison between r and d. That is what we are looking for.

Because the first statement makes a direct, numerical comparison between r and d, it is sufficient by itself to answer the question.

Let's look at the second statement.

Translating that into math, we get:

$4d<2\pi r$ ,

where d is the length of the square's side and r is the radius of the circle.

This also sets an equivalence (technically, an inequality) between d and r, and we could use that information to show whether the area of the circle is less than the area of the square.

Statement 2 is ALSO sufficient to answer the question.

So, the answer choice is D. Both statements are sufficient on their own to answer the question.

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Question

Two circles are constructed; one is inscribed inside a given equilateral triangle, and the other is circumscribed about the same triangle.

The circumscribed circle has circumference $20 \pi$ . Give the area of the inscribed circle.

Answer

Examine the diagram below, which shows the triangle, its three altitudes, and the two circles.

The three altitudes of an equilateral triangle divide one another into two segments each, the longer of which is twice the length of the shorter. The length of each of the longer segments is the radius of the circumscribed circle, and the length of each of the shorter segments is the radius of the inscribed circle. Therefore, the inscribed circle has half the radius of the circumscribed circle.

The circumscribed circle has circumference $20 \pi$ , so its radius is

$20 \pi \div 2 \pi = 10$

The inscribed circle has radius half this, or 5, so its area is

$A= \pi r^{2} = \pi \cdot 5^{2}= 25 \pi$

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Question

Two circles are constructed; one is inscribed inside a given equilateral triangle, and the other is circumscribed about the same triangle.

The inscribed circle has circumference $20 \pi$ . Give the area of the circumscribed circle.

Answer

Examine the diagram below, which shows the triangle, its three altitudes, and the two circles.

The three altitudes of an equilateral triangle divide one another into two segments each, the longer of which is twice the length of the shorter. The length of each of the longer segments is the radius of the circumscribed circle, and the length of each of the shorter segments is the radius of the inscribed circle. Therefore, the circumscribed circle has twice the radius of the inscribed circle.

The inscribed circle has circumference $20 \pi$ , so its radius is

$20 \pi \div 2 \pi = 10$

The circumscribed circle has radius twice this, or 20, so its area is

$A= \pi r^{2} = \pi \cdot 20^{2}= 400 \pi$

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Question

Two circles are constructed; one is inscribed inside a given square, and the other is circumscribed about the same square.

The inscribed circle has circumference $20 \pi$ . Give the area of the circumscribed circle.

Answer

Examine the diagram below, which shows the square, segments from its center to a vertex and the midpoint of a side, and the two circles.

Note that the segment from the center of the square to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two segments and half a side of the square form a 45-45-90 triangle, so by the 45-45-90 Theorem, the radius of the circumscribed circle is $\sqrt{2}$ times that of the inscribed circle.

The inscribed circle has circumference $20 \pi$ , so its radius is

$20 \pi \div 2 \pi = 10$

The circumscribed circle has radius $\sqrt{2}$ times this, or $10 \sqrt{2}$ , so its area is

$A= \pi r^{2} = \pi \cdot \left (10 \sqrt{2} \right ) ^{2}= \pi \cdot 100 \cdot 2 = 200 \pi$

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Question

Two circles are constructed; one is inscribed inside a given square, and the other is circumscribed about the same square.

The circumscribed circle has circumference $20 \pi$ . Give the area of the inscribed circle.

Answer

Examine the diagram below, which shows the square, segments from its center to a vertex and the midpoint of a side, and the two circles.

Note that the segment from the center of the square to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two radii and half a side of the square form a 45-45-90 Triangle, so by the 45-45-90 Theorem, the radius of the inscribed circle is equal to that of the circumscribed circle divided by $\sqrt{2}$ .

The inscribed circle has circumference $20 \pi$ , so its radius is

$20 \pi \div 2 \pi = 10$

Divide this by $\sqrt{2}$ to get the radius of the circumscribed circle:

$\frac{10}{\sqrt{2}}= \frac{10 \cdot \sqrt{2}}{\sqrt{2} \cdot \sqrt{2}} = \frac{10 \sqrt{2}}{2} = 5 \sqrt{2}$

The circumscribed circle has area

$A= \pi r^{2} = \pi \cdot \left (5 \sqrt{2} \right ) ^{2}= \pi \cdot 25 \cdot 2 = 50 \pi$

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Question

Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.

The inscribed circle has circumference $20 \pi$ . Give the area of the circumscribed circle.

Answer

Examine the diagram below, which shows the hexagon, segments from its center to a vertex and the midpoint of a side, and the two circles.

Note that the segment from the center of the hexagon to the midpoint of a side is a radius of the inscribed circle, and the segment from the center to a vertex is a radius of the circumscribed circle. The two segments and half a side of the hexagon can be proved to form a 30-60-90 triangle.

The inscribed circle has circumference $20 \pi$ , so its radius - and the length of the longer leg of the right triangle - is

$20 \pi \div 2 \pi = 10$

By the 30-60-90 Theorem, the length of the shorter leg is this length divided by $\sqrt{3}$ , or $\frac{10}{\sqrt{3}}$ ; the length of the hypotenuse, which is the radius of the circumscribed circle, is twice this, or $\frac{20}{\sqrt{3}}$ .

The area of the circumscribed circle can now be calculated:

$A= \pi r^{2} = \pi \cdot \left (\frac{20}{\sqrt{3}}\right ) ^{2}= \pi \cdot \frac{400}{3} = \frac{400 \pi }{3}$

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