Diameter - GMAT Quantitative

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Question

Find the center of the circle that has a diameter with endpoints at and .

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Answer

$midpoint = (\frac{x_{1}+x_{2}}{2},\frac{y_{1}+y_{2}}{2} ) = (\frac{-4 + 0}{2}, \frac{4 + 2}{2}) = (\frac{-4}{2}, \frac{6}{2}) = (-2, 3)$

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Question

A square is inscribed inside a circle. The square has an area of 100. Find the area of the circle.

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Answer

Since the square is inscribed inside the circle, then the length of the diagonal of the square will be the diameter of the circle. The area of a circle is given as $A=\pi r^2$ and

$\frac{diameter}{2}=radius \Rightarrow r=\frac{d}{2}.$

Substituting the diameter into the equation of the area we get

$A=\pir^2=\pi\left (\frac{d}{2}\right )^2=\frac{\pid^2}{4}.$

So if we find the diagonal of the square, then we can find the area of the circle. To find the diagonal of the square we use the fact that the area of a square is where is the side of the square. Since the area is 100, then the length of the square is 10. The diagonal is then given by

$diagonal=\sqrt{10^2+10^2}=\sqrt{200}=10\sqrt{2}.$

Substituting this into our equation for the area of a circle, we get

$A=\frac{\pi(10\sqrt{2})^2}{4}=\frac{\pi1002}{4}=\pi252=50\pi$

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Question

Consider the Circle :

(Figure not drawn to scale.)

If is the center of the circle and is a point on its circumference, what is the length of the diameter of the circle?

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Answer

This problem provides you with a circle and gives you clues that line is the radius. A line that passes through the center and ends on the circumference of the circle is the radius of the circle. In this case, our radius is 15 meters. Diameter is simply twice the radius, so our diameter is 30 meters.

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Question

A circle has a circumference of $16\pi:cm$ . What is the diameter of the circle?

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Answer

We can find the diameter of the circle using the formula for circumference:

$C=2\pi r$

First, we must recognize that the radius is just half of the diameter, which allows us to express the formula in terms of diameter:

$C=2\pi r=2\pi (\frac{d}{2})=\pi d$

Now we can just plug in the given circumference and solve for the diameter:

$C=\pi d\rightarrow 16\pi:cm=\pi d\rightarrow d=16:cm$

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Question

If the area of a circle is $9\pi:cm^2$ , what is the diameter of the circle?

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Answer

We are given the area of the circle, so we will need to use the formula for area to calculate its diameter:

$A=\pi r^2$

First, we must express the formula in terms of diameter instead of radius. The radius is just half of the diameter, so we can write:

$A=\pi r^2=\pi (\frac{d}{2})^2=\frac{\pi }{4}d^2$

Now we can simply plug in the given area and solve for the diameter:

$A=\frac{\pi }{4}d^2\rightarrow 9\pi:cm^2=\frac{\pi }{4}d^2\rightarrow d^2=36:cm^2\rightarrow d=6:cm$

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Question

A given circle has an area of $81\pi$ . What is the length of its diameter?

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Answer

The area of a circle is defined by the equation $A=\pi r^{2}$ , where is the length of the circle's radius. The radius, in turn, is defined by the equation $r=\frac{d}{2}$ , where is the length of the circle's diameter.

Given $A=81\pi$ , we can deduce that $r^{2}=81$ and therefore . Then, since $r=\frac{d}{2}$ , .

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Question

A given circle has a radius of . What is the diameter of the circle?

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Answer

By definition, the length of a circle's diameter is twice the length of the circle's radius , or . Since , .

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Question

The area of a given circle is $10\pi$ . What is the diameter of the circle?

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Answer

The area of a circle is defined by the equation $A=\pi r^{2}$ , where is the length of the circle's radius. The radius, in turn, is defined by the equation $r=\frac{d}{2}$ , where is the length of the circle's diameter.

Given $A=10\pi$ , we can deduce that $r^{2}=10$ and therefore $r=\sqrt{10}$ . Then, since $r=\frac{d}{2}$ , $d=2r=2\sqrt{10}$ .

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Question

While SCUBA diving, Dirk uncovers a strange circular rock formation. He estimates the area of the circular formation to be . Help Dirk find the diameter of the surface.

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Answer

While SCUBA diving, Dirk uncovers a strange circular rock formation. He estimates the area of the circular formation to be . Help Dirk find the diameter of the surface.

Recall the formula for area of a circle:

$A=\pi r^2$

Since we know A, we can solve for r

$144=\pi r^2$

Rearranging and simplifing gets our radius to be:

$r\approx6.77m$

We are almost there, but we have one more step to go. We need to double the radius to get the diameter...

$d\approx13.54m$

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Question

What is the diameter of a circle with an area of $100\pi units^{2}$

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Answer

The area of a circle can be represented by the equation:

$A = \pi r^2$

Therefore:

$\frac{100\pi}{\pi} = \frac{\pi r^2}{\pi} = 100 = r^2$

$\sqrt{r^2} = \sqrt{100} = r = 10$

Since the diameter is twice the radius,

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Question

If an ecologist is measuring species composition using a circular region with an area of , what is the diameter of the region?

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Answer

If an ecologist is measuring species composition using a circular region with an area of , what is the diameter of the region?

Recall the formula for the area of a circle:

$A=\pi r^2$

We also know that r, the radius, is half the length of the diameter. Therefore, if we can find the radius, we can find the diameter:

$3.75=\pi r^2$

$3.75\div \pi =r^2$

$r=\sqrt{1.19}\approx1.09yd$

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Question

For any circle, what is the ratio of its circumference to its diameter?

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Answer

In order to calculate the ratio of circumference to diameter, we need an equation that involves both variables. The formula for circumference is as follows:

$C=2\pi r$

We need to express the radius in terms of diameter. The radius of a circle is half of its diameter, so we can rewrite the formula as:

$C=2\pi r=2\pi (\frac{d}{2})=\pi d$

If we divide both sides by the diameter, on the left side we will have $\frac{C}{d}$ , which is the ratio of circumference to diameter:

$C=\pi d$

$\frac{C}{d}=\pi$

So, for any circle, the ratio of its circumference to its diameter is equal to $\pi$ , which is actually the definition of this very important mathematical constant.

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Question

For any given circle, what is the ratio of its diameter to its circumference?

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Answer

To find the ratio between the diameter and the circumference of a circle, we need to use the formula for the circumference of a circle:

$C=2\pi r$

We can see this formula is in terms of radius, so we need to express it in a way that the circumference is in terms of diameter. Using the knowledge that the radius is half of the diameter:

$C=2\pi r\rightarrow C=2\pi (\frac{d}{2})\rightarrow C=\pi d$

Now that we have a simple formula involving the circumference and diameter, we can see that we will have the ratio of diameter to circumference if we divide both sides by the circumference. We then divide both sides by $\pi$ to isolate the ratio of diameter to circumference and find our solution:

$C=\pi d$

$\frac{d}{C}=\frac{1}{\pi }$

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Question

The circle with center , is inscribed in the square . What is the ratio of the diameter to the circumference of the circle given that the square has an area of ?

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Answer

To calculate the ratio of the diameter to the circumference of the square we should first get the diameter, which is the same as the length of a side of the sqaure. To do so we just need to take the square root of the area of the square, which is 4. Also we should remember that the circumference is given by $2\pi r$ , where is the length of the radius.

Now we should notice that this formula can also be written $d\pi$ .

The ratio we are looking for is $\frac{d}{d\pi}=\frac{1}{\pi}$ . Therefore, this ratio will always be $\frac{4}{4\pi}=\frac{1}{\pi}$ and this is our final answer.

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Question

Equilateral triangle is inscribed in a circle. The perimeter of the triangle is . What is the radius of the circle?

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Answer

The perimeter of the circle allows us to find the side of equilateral triangle ABC; $\frac{6}{3}$ or 2.

From there, we can also find the length of the height of equilateral triangle ABC with the formula $h=\frac{s\sqrt{3}}{2}$ , which turns out to be $\sqrt{3}$ .

Since in an equilateral triangle, the center of gravity is at $\frac{2}{3}$ from any vertex, it follows that the radius of the circle is $\frac{2\sqrt{3}}{3}$ .

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Question

A given circle has a circumference of $6\pi$ and a radius of . What is the ratio of the circle's circumference to its diameter?

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Answer

For a given circle of circumference and radius , $C=2\pi r$ .

Since the radius of a circle is equal to half of the circle's diameter , we can then define as

$C=2\pi\left(\frac{d}{2}\right)=\pi d$ .

Therefore, the ratio of the circumference to the diameter of this and all other circles is $\frac{C}{d}=\frac{6\pi}{3\cdot2}=\frac{6\pi}{6}=\pi$ .

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Question

A given circle has a circumference of $10\pi$ and a radius of . What is the ratio of the circle's circumference to its diameter?

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Answer

For a given circle of circumference and radius , $C=2\pi r$ .

Since the radius of a circle is equal to half of the circle's diameter , we can then define as

$C=2\pi\left(\frac{d}{2}\right)=\pi d$ .

Therefore, the ratio of the circumference to the diameter of this and all other circles is,

$\frac{C}{d}=\frac{10\pi}{5\cdot2}=\frac{10\pi}{10}=\pi$ .

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Question

A given circle has a circumference of $14\pi$ and a radius of . What is the ratio of the circle's circumference to its diameter?

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Answer

For a given circle of circumference and radius , $C=2\pi r$ .

Since the radius of a circle is equal to half of the circle's diameter , we can then define as,

$C=2\pi(\frac{d}{2})=\pi d$ .

Therefore, the ratio of the circumference to the diameter of this circles is,

$\frac{C}{d}=\frac{14\pi}{7\cdot2}=\frac{14\pi}{14}=\pi$ .

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Question

Find the circumference of a circle with a diameter measuring $17\textup{ in}$ .

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Answer

The circumference of a circle is given by $C=2 r \pi =d\pi$

where $\pi = 3.14$

We are told the diameter so we just need to plug in our value into the equation:

$C=(17)\pi = 17 \cdot 3.14 = 53.38$

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Question

Find the circumference of a circle with a diameter measuring $17\textup{ in}$ .

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Answer

The circumference of a circle is given by $C=2 r \pi =d\pi$

where $\pi = 3.14$

We are told the diameter so we just need to plug in our value into the equation:

$C=(17)\pi = 17 \cdot 3.14 = 53.38$

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