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.

The formula for finding the area of a circle is \(\displaystyle \pi r^{2}\). In this formula, \(\displaystyle r\) represents the radius of the circle.  Since the question only gives us the measurement of the diameter of the circle, we must calculate the radius.  In order to do this, we divide the diameter by \(\displaystyle 2\).

\(\displaystyle \frac{15}{2}=7.5\)

Now we use \(\displaystyle 7.5\) for \(\displaystyle r\) in our equation.

\(\displaystyle \pi (7.5)^{2}=176.7146 \: in^{2}\)

First, solve for radius:

\(\displaystyle r=\frac{d}{2}=\frac{6}{2}=3\)

Then, solve for area:

\(\displaystyle A=r^2\pi=3^2\pi=9\pi\)

The area of a circle can be calculated using the formula:

\(\displaystyle Area=\frac{\pi d^2}{4}\),

where \(\displaystyle d\) is the diameter of the circle, and \(\displaystyle \pi\) is approximately \(\displaystyle 3.14\).

\(\displaystyle Area=\frac{\pi d^2}{4}=\frac{\pi\times 4^2}{4}=4\pi \Rightarrow Area\approx 4\times 3.14\Rightarrow Area\approx 12.56 \ cm^2\)

The area of a circle can be calculated using the formula:

\(\displaystyle Area=\frac{\pi d^2}{4}\),

where \(\displaystyle d\)  is the diameter of the circle and \(\displaystyle \pi\) is approximately \(\displaystyle 3.14\).

\(\displaystyle Area=\frac{\pi (4t)^2}{4}=\frac{16\pi t^2}{4}=4\pi t^2 \Rightarrow Area\approx 4\times 3.14\times t^2\)

\(\displaystyle \Rightarrow Area\approx 12.56t^2\)