GMAT Quantitative - Diameter

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

Not enough information provided

Explanation

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 $r=\frac{d}{2}$ , .

Consider the Circle :

Circle3

(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?

Explanation

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.

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.

Explanation

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...

$d\approx13.54m$

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

$34cm^{2}$

$\frac{17}{2}cm$

Explanation

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

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

$50\pi$

$25\pi$

$75\pi$

$12.5\pi$

Explanation

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

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

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

$A=\frac{\pi*(10\sqrt{2})^2}{4}=\frac{\pi*100*2}{4}=\pi*25*2=50\pi$

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

$2\sqrt{10}$

$\sqrt{10}$

Not enough information provided

Explanation

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

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

Explanation

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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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 ?

$\frac{1}{\pi}$

$\frac{4}{\pi}$

$\pi$

$\sqrt{2}$

Explanation

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.

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

$\pi$

$2\pi$

$\frac{\pi }{2}$

$\frac{\pi }{4}$

$4\pi$

Explanation

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.