# ISEE Upper Level Math : Circles

## Example Questions

### Example Question #1 : Radius

What is the radius of a circle with circumference equal to ?

Explanation:

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

### Example Question #2 : How To Find The Length Of A Radius

What is the value of the radius of a circle if the area is equal to ?

Explanation:

The equation for finding the area of a circle is

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

### Example Question #3 : How To Find The Length Of A Radius

What is the radius of a circle with a circumference of ?

Explanation:

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

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

### Example Question #4 : How To Find The Length Of A Radius

Refer to the above diagram.  has length . Give the radius of the circle.

Explanation:

Inscribed , which measures , intercepts a minor arc with twice its measure. That arc is , which consequently has measure

.

The corresponding major arc, , has as its measure

, and is

of the circle.

If we let  be the circumference and  be the radius, then  has length

.

This is equal to , so we can solve for  in the equation

The radius of the circle is 50.

### Example Question #1 : Radius

A circle has a circumference of . What is the radius of the circle?

Not enough information to determine.

Explanation:

A circle has a circumference of . What is the radius of the circle?

Begin with the formula for circumference of a circle:

Now, plug in our known and work backwards:

Divide both sides by two pi to get:

### Example Question #6 : How To Find The Length Of A Radius

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 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 .

What is the radius 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.

So our answer is 13m.

### Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

What is the area of a circle that has a diameter of inches?

Explanation:

The formula for finding the area of a circle is . In this formula, 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 .

Now we use for in our equation.

### Example Question #1 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

What is the area of a circle with a diameter equal to 6?

Explanation:

First, solve for radius:

Then, solve for area:

### Example Question #2 : Know And Use The Formulas For The Area And Circumference Of A Circle: Ccss.Math.Content.7.G.B.4

The diameter of a circle is . Give the area of the circle.

Explanation:

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

,

where is the diameter of the circle, and is approximately .

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

The diameter of a circle is . Give the area of the circle in terms of .