To find the circumference of a circle, we will use the following formula:

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

where r is the radius of the circle. 

Now, we know the radius of the circle is 8in.

We also know that\(\displaystyle \pi = 3.14\).

Knowing all of this, we can substitute into the formula.  We get

\(\displaystyle C = 2 \cdot 3.14 \cdot 8\text{in}\)

\(\displaystyle C = 16\text{in} \cdot 3.14\)

\(\displaystyle C = 50.24\text{in}\)

You want to ice skate around the outer edge of a circular pond (the water is frozen, because it is January and you live in Wisconsin). If the pond has a radius of 35 meters, how far will you skate in one lap around the pond?

Let's begin by realizing what we are being asked for. They want us to find the total distance we will travel in one lap around the pond. This sounds like circumference, because we are going one full distance around the outside of the circle.

The formula for circumference is as follows.

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

Now, we have r, so just plug in and solve.

\(\displaystyle C=2 \pi (35m)=2*35 m \pi=70 \pi m\)

So, our answer is:

\(\displaystyle 70 \pi m\)

You are conducting fieldwork, when you find a tree whose radius at chest height is \(\displaystyle 1.5m\). What is the circumference of the tree at chest height?

The formula for circumference of a circle is:

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

We have r, so simply plug it in and solve:

\(\displaystyle C=2 \pi (1.5m)=3 \pi m\)

So, our answer is \(\displaystyle 3 \pi m\)

To find the circumference of a circle, we will use the following formula:

\(\displaystyle C = \pi d\)

where d is the diameter of the circle.

Now, we know the diameter of the circle is 18cm.  So, we will substitute.  We get

\(\displaystyle C = \pi \cdot 18\text{cm}\)

\(\displaystyle C = 18\pi \text{ cm}\)

To find the circumference of a circle, we will use the following formula:

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

where r is the radius of the circle. 

Now, we know the radius of the circle is 10in.  

So, we get

\(\displaystyle C = 2 \cdot \pi \cdot 10\text{in}\)

\(\displaystyle C = 20\text{in} \cdot \pi\)

\(\displaystyle C = 20\pi \text{ in}\)

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 circumference 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.

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

Next, plug our answer into the circumference formula.

\(\displaystyle C=2 \pi r = 2 (13m) \pi =26 \pi m\)

Sector TYP occupies 43% of a circle. Find the degree measure of angle TYP.

Use the following formula and solve for x:

\(\displaystyle \frac{x}{360^{\circ}}100=43\%\)

Begin by dividing over the 100

\(\displaystyle \frac{x}{360^{\circ}}=.43\)

Then multiply by 360

\(\displaystyle x=.43*360^{\circ}=154.8^{\circ}\)

If sector AJL covers 45% of circle J, what is the measure of sector AJL's central angle?

To find an angle measure from a percentage, simply convert the percentage to a decimal and then multiply it by 360 degrees.

\(\displaystyle 45\%\rightarrow 0.45\)

\(\displaystyle 0.45 * 360^{\circ}=162^{\circ}\)

So, our answer is 162 degrees.

If we let \(\displaystyle C\) be the circumference of the circle, then the length of \(\displaystyle \overarc{AB}\) is \(\displaystyle \frac{80}{360} = \frac{2}{9 }\) of the circumference, so

\(\displaystyle \frac{2}{9 } C = 12 \pi\)

\(\displaystyle \frac{9 }{2}\cdot \frac{2}{9 } C =\frac{9 }{2}\cdot 12 \pi\)

\(\displaystyle C =54\pi\)

The radius is the circumference divided by \(\displaystyle 2 \pi\):

\(\displaystyle r= 54 \pi \div 2 \pi = 27\)

Use the formula to find the area of the entire circle:

\(\displaystyle A = \pi r ^{2} = \pi \left (27 \right )^{2} = 729 \pi\)

The area of the white region is \(\displaystyle 1 - \frac{2}{9} = \frac{7}{9}\) of that of the circle, or 

\(\displaystyle \frac{7}{9} \cdot 729 \pi = 567 \pi\)

The radius of a circle is found by dividing the circumference \(\displaystyle C = 30 \pi\) by \(\displaystyle 2 \pi\):

\(\displaystyle r= \frac{C}{2 \pi}= \frac{30 \pi}{2 \pi} = 15\)

The area of the entire circle can be found by substituting for \(\displaystyle r\) in the formula:

\(\displaystyle A = \pi r ^{2} = \pi \cdot 15 ^{2} = 225 \pi\).

The area of the shaded \(\displaystyle 80 ^{\circ }\) sector is \(\displaystyle \frac{80 }{360 }\) of the total area:

\(\displaystyle \frac{80 }{360 } \times 225 \pi = \frac{2}{9} \times 225 \pi = 50 \pi\)