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

\(\displaystyle C=d\pi=2r\pi\)

\(\displaystyle 36\pi=2r\pi\)

\(\displaystyle \frac{36\pi}{\pi}=\frac{2r\pi}{\pi}\)

\(\displaystyle 36=2r\)

\(\displaystyle \frac{36}{2}=\frac{2r}{2}\)

\(\displaystyle 18=r\)

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

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

\(\displaystyle \pi r^{2}=18\pi\)

\(\displaystyle r^{2}=18\)

\(\displaystyle r=\sqrt{18}=\sqrt{9\cdot 2}=3\sqrt{2}\)

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

\(\displaystyle C=2r\pi\)

We plug in the circumference given, \(\displaystyle 24\pi\) into \(\displaystyle C\) and use algebraic operations to solve for \(\displaystyle r\).

\(\displaystyle 24\pi=2r\pi\)

\(\displaystyle \frac{24\pi }{\pi }=\frac{2r\pi }{\pi }\)

\(\displaystyle 24=2r\)

\(\displaystyle \frac{24}{2}=\frac{2r}{2}\)

\(\displaystyle r=12\)

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

\(\displaystyle 72 ^{\circ } \times 2 = 144 ^{\circ }\).

The corresponding major arc, \(\displaystyle \overarc {ACB}\), has as its measure

\(\displaystyle 360 ^{\circ } - 144 ^{\circ } = 216 ^{\circ }\), and is

\(\displaystyle \frac{ 216 }{360 } = \frac{ 216 \div 72 }{360 \div 72 } = \frac{3}{5}\)

of the circle.

If we let \(\displaystyle C\) be the circumference and \(\displaystyle r\) be the radius, then \(\displaystyle \overarc {ACB}\) has length

\(\displaystyle \frac{3}{5} C = \frac{3}{5} \cdot 2 \pi r = \frac{6 \pi}{5} r\).

This is equal to \(\displaystyle 60 \pi\), so we can solve for \(\displaystyle r\) in the equation

\(\displaystyle \frac{6}{5} \pi r = 60 \pi\)

\(\displaystyle \frac{5}{6 \pi } \cdot \frac{6 \pi }{5} r = \frac{5}{6 \pi } \cdot 60 \pi\)

\(\displaystyle r = 50\)

The radius of the circle is 50.

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

Begin with the formula for circumference of a circle:

\(\displaystyle Circumference=2\pi r\)

Now, plug in our known and work backwards:

\(\displaystyle 768\pi m=2\pi r\)

Divide both sides by two pi to get:

\(\displaystyle 384m=r\)

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 \(\displaystyle 169 \pi m^2\).

What is the radius of the crater?

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

\(\displaystyle A=\pi r^2\)

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

\(\displaystyle 169 \pi m^2=\pi r ^2\)

Begin by dividing out the pi

\(\displaystyle 169m^2=r^2\)

Then, square root both sides.

\(\displaystyle r=\sqrt{169m^2}=13m\)

So our answer is 13m.