Calculating the area of a circle - GMAT Quantitative
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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?
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?
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For a circle, 
.
Therefore, the area of the 12" pizza 
.
The area of the two 6" pizzas 
.
The area of the three 4" pizzas = 
.
The 12" pizza is the best option.
For a circle,
Therefore, the area of the 12" pizza
The area of the two 6" pizzas
The area of the three 4" pizzas =
The 12" pizza is the best option.
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A circle is inscribed in a square with area 100. What is the area of the circle?
A circle is inscribed in a square with area 100. What is the area of the circle?
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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 
, so the answer is 
.
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
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What is the area of a circle with a diameter of 
?
What is the area of a circle with a diameter of
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The area 
of a circle is defined by 
, where 
is the radius of the circle. We are provided with the diameter 
of the circle, which is twice the length of 
.
If 
, then 
Then, solving for 
:
The area
If
Then, solving for
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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?
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?
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The largest square you could make would be 
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
Then the area will be
This is closer to 280 than to 300
The largest square you could make would be
Then the area will be
This is closer to 280 than to 300
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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?
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?
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Let 
be the radius of the circle. Its area is 
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: 
This makes the ratio of the area of the circle to that of the square 
.
Let
The diagonal of the square is equal to the diameter of the circle, or
This makes the ratio of the area of the circle to that of the square
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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 
?
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
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The area of a circle with radius 12 is
.
This is also the area of the square, so the sidelength of that square is the square root of the area:
The area of a circle with radius 12 is
This is also the area of the square, so the sidelength of that square is the square root of the area:
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What is the area of a circle which goes through the points 
?
What is the area of a circle which goes through the points
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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 
, angle intercepts a 
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:
The radius of the circle is half this, or 
.
The area of the circle is therefore:
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
The radius of the circle is half this, or
The area of the circle is therefore:
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A circle on the coordinate plane has equation 
.
What is its area?
A circle on the coordinate plane has equation
What is its area?
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The equation of a circle centered at the origin is
,
where 
is the radius of the circle.
The area of a circle is 
.
From the equation given in the question stem, we know that 
, so we can plug this into the area formula:
The equation of a circle centered at the origin is
where
The area of a circle is
From the equation given in the question stem, we know that
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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.
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.
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The larger circle has diameter one yard, or 36 inches; its radius is half that, or 18 inches, so its area is
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:
square inches.
The area of the gray region is the difference of the two:
square inches.
The larger circle has diameter one yard, or 36 inches; its radius is half that, or 18 inches, so its area is
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:
The area of the gray region is the difference of the two:
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Refer to the above circle. The smaller circle has diameter 
feet. In terms of 
, give the area of the gray region in square inches.
Refer to the above circle. The smaller circle has diameter
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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 
, we can substitute these quantities for 
and subtract the area of the smaller circle from that of the larger:
square inches
square inches
square inches
Multiply the diameter of the smaller circle by 12 to convert it to inches - this will be
Using the area formula for a circle
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A circle on the coordinate plane has equation
Which of the following represents its area?
A circle on the coordinate plane has equation
Which of the following represents its area?
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The equation of a circle centered at the origin is
where 
is the radius of the circle.
The area of a circle is 
; since in the equation of the circle, 
, we can substitute to get the area 
.
The equation of a circle centered at the origin is
where
The area of a circle is
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A circle on the coordinate plane has equation
What is its area?
A circle on the coordinate plane has equation
What is its area?
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The area of a circle is equal to 
, where 
is the radius.
The standard form of the area of a circle with radius 
and center 
is
Once we get the equation in standard form, we know 
, which can be multiplied by 
.
Complete the squares:
so 
can be rewritten as follows:
Therefore, 
and 
.
The area of a circle is equal to
The standard form of the area of a circle with radius
Once we get the equation in standard form, we know
Complete the squares:
so
Therefore,
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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
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
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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:
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:
,
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.
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:
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:
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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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 
. Give the area of the inscribed circle.
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
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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 
, so its radius is
The inscribed circle has radius half this, or 5, so its area is
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
The inscribed circle has radius half this, or 5, so its area is
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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 
. Give the area of the circumscribed circle.
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
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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 
, so its radius is
The circumscribed circle has radius twice this, or 20, so its area is
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
The circumscribed circle has radius twice this, or 20, so its area is
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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 
. Give the area of the circumscribed circle.
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
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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 
times that of the inscribed circle.
The inscribed circle has circumference , so its radius is
The circumscribed circle has radius 
times this, or 
, so its area is
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
The inscribed circle has circumference
The circumscribed circle has radius
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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 
. Give the area of the inscribed circle.
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
Tap to reveal 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 
.
The inscribed circle has circumference , so its radius is
Divide this by 
to get the radius of the circumscribed circle:
The circumscribed circle has area
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
The inscribed circle has circumference
Divide this by
The circumscribed circle has area
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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 
. Give the area of the circumscribed circle.
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
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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 , so its radius - and the length of the longer leg of the right triangle - is
By the 30-60-90 Theorem, the length of the shorter leg is this length divided by 
, or 
; the length of the hypotenuse, which is the radius of the circumscribed circle, is twice this, or 
.
The area of the circumscribed circle can now be calculated:
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
By the 30-60-90 Theorem, the length of the shorter leg is this length divided by
The area of the circumscribed circle can now be calculated:
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Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.
The circumscribed circle has circumference 
. Give the area of the inscribed circle.
Two circles are constructed; one is inscribed inside a given regular hexagon, and the other is circumscribed about the same hexagon.
The circumscribed circle has circumference
Tap to reveal 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 circumscribed circle has circumference , so its radius - and the length of the hypotenuse of the right triangle -
By the 30-60-90 Theorem, the length of the shorter leg is half this, or 5. The length of the longer leg, which is the radius of the inscribed circle, is 
times this, or 
.
The area of the inscribed circle can now be calculated:
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 circumscribed circle has circumference
By the 30-60-90 Theorem, the length of the shorter leg is half this, or 5. The length of the longer leg, which is the radius of the inscribed circle, is
The area of the inscribed circle can now be calculated:
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In the figure shown below, line segment 
passes through the center of the circle and has a length of 
. Points 
, 
, and 
are on the circle. Sector 
covers 
of the total area of the circle. Answer the following questions regarding this shape.
Find the area of the given shape.
In the figure shown below, line segment
Find the area of the given shape.
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Finding the area here is straightforward if we can use what we're given wisely. First, the area of a circle can be found using the following:
where 
is area and 
is our radius. Now, although we are not given our radius, we are given the diameter, just not by name.
Segment 
goes from one point on the outside of the circle through the center to another point on the circle; that means its length is the circle's diameter.
If our diameter is 
, then our radius must be 
. Plug this into the area equation to find the area of our circle:
Finding the area here is straightforward if we can use what we're given wisely. First, the area of a circle can be found using the following:
where
Segment
If our diameter is
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