Calculating the length of the diameter - GMAT Quantitative Reasoning

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Question

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

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

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

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.

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?

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?

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?

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?

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?

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

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.

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 $100\pi units^{2}$

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?

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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