# ISEE Upper Level Math : Circles

## Example Questions

### Example Question #12 : How To Find Circumference

Let

Find the circumference of a circle with a radius of 8in.

Explanation:

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

where r is the radius of the circle.

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

We also know that.

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

### Example Question #13 : How To Find Circumference

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?

Not enough information to complete.

Explanation:

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.

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

### Example Question #14 : How To Find Circumference

You are conducting fieldwork, when you find a tree whose radius at chest height is . What is the circumference of the tree at chest height?

Explanation:

You are conducting fieldwork, when you find a tree whose radius at chest height is . What is the circumference of the tree at chest height?

The formula for circumference of a circle is:

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

### Example Question #15 : How To Find Circumference

Find the circumference of a circle with a diameter of 18cm.

Explanation:

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

where d is the diameter of the circle.

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

### Example Question #16 : How To Find Circumference

Find the circumference of a circle with a radius of 10in.

Explanation:

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

where r is the radius of the circle.

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

So, we get

### Example Question #17 : How To Find Circumference

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 .

What is the circumference of the crater?

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 .

What is the circumference of the crater?

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

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

Begin by dividing out the pi

Then, square root both sides.

Next, plug our answer into the circumference formula.

### Example Question #1 : How To Find The Angle For A Percentage Of A Circle

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

Explanation:

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

Use the following formula and solve for x:

Begin by dividing over the 100

Then multiply by 360

### Example Question #2 : How To Find The Angle For A Percentage Of A Circle

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

Explanation:

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.

So, our answer is 162 degrees.

### Example Question #1 : How To Find The Area Of A Sector

Give the area of the white region of the above circle if  has length

Explanation:

If we let  be the circumference of the circle, then the length of  is  of the circumference, so

The radius is the circumference divided by :

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

The area of the white region is  of that of the circle, or

### Example Question #2 : How To Find The Area Of A Sector

The circumference of the above circle is . Give the area of the shaded region.