How to find the length of a radius

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ISEE Upper Level Quantitative Reasoning › How to find the length of a radius

Questions 1 - 6

1

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the radius of the crater?

Cannot be determined from the information provided

Explanation

You are exploring the woods near your house, when you come across an impact crater. It is perfectly circular, and you estimate its area to be $169 \pi m^2$ .

What is the radius of the crater?

To solve this, we need to recall the formula for the area of a circle.

$A=\pi r^2$

Now, we know A, so we just need to plug in and solve for r!

$169 \pi m^2=\pi r ^2$

Begin by dividing out the pi

Then, square root both sides.

$r=\sqrt{169m^2}=13m$

So our answer is 13m.

2

Inscribed angle

Refer to the above diagram. $\overarc {ACB}$ has length $60 \pi$ . Give the radius of the circle.

Explanation

Inscribed $\angle ACB$ , which measures $72 ^{\circ }$ , intercepts a minor arc with twice its measure. That arc is $\overarc {AB}$ , which consequently has measure

$72 ^{\circ } \times 2 = 144 ^{\circ }$ .

The corresponding major arc, $\overarc {ACB}$ , has as its measure

$360 ^{\circ } - 144 ^{\circ } = 216 ^{\circ }$ , and is

$\frac{ 216 }{360 } = \frac{ 216 \div 72 }{360 \div 72 } = \frac{3}{5}$

of the circle.

If we let be the circumference and be the radius, then $\overarc {ACB}$ has length

$\frac{3}{5} C = \frac{3}{5} \cdot 2 \pi r = \frac{6 \pi}{5} r$ .

This is equal to $60 \pi$ , so we can solve for in the equation

$\frac{6}{5} \pi r = 60 \pi$

$\frac{5}{6 \pi } \cdot \frac{6 \pi }{5} r = \frac{5}{6 \pi } \cdot 60 \pi$

The radius of the circle is 50.

3

A circle has a circumference of $768\pi m$ . What is the radius of the circle?

Not enough information to determine.

Explanation

A circle has a circumference of $768\pi m$ . What is the radius of the circle?

Begin with the formula for circumference of a circle:

$Circumference=2\pi r$

Now, plug in our known and work backwards:

$768\pi m=2\pi r$

Divide both sides by two pi to get:

4

What is the radius of a circle with circumference equal to $36\pi$ ?

Explanation

The circumference of a circle can be found using the following equation:

$C=d\pi=2r\pi$

$36\pi=2r\pi$

$\frac{36\pi}{\pi}=\frac{2r\pi}{\pi}$

$\frac{36}{2}=\frac{2r}{2}$

5

What is the value of the radius of a circle if the area is equal to $18\pi$ ?

$3\sqrt{2}$

$2\sqrt{3}$

Explanation

The equation for finding the area of a circle is $\pi r^{2}$ .

Therefore, the equation for finding the value of the radius in the circle with an area of $18\pi$ is:

$\pi r^{2}=18\pi$

$r^{2}=18$

$r=\sqrt{18}=\sqrt{9\cdot 2}=3\sqrt{2}$

6

What is the radius of a circle with a circumference of $24\pi$ ?

Explanation

The circumference of a circle can be found using the following equation:

$C=2r\pi$

We plug in the circumference given, $24\pi$ into and use algebraic operations to solve for .

$24\pi=2r\pi$

$\frac{24\pi }{\pi }=\frac{2r\pi }{\pi }$

$\frac{24}{2}=\frac{2r}{2}$

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