# ISEE Upper Level Math : Circles

## Example Questions

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

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

Explanation:

The area of a circle can be calculated as , where   is the radius of the circle, and is approximately .

### Example Question #12 : Geometry

The perpendicular distance from the chord to the center of a circle is , and the chord length is . Give the area of the circle in terms of .

Explanation:

Chord length = , where   is the radius of the circle and   is the perpendicular distance from the chord to the circle center.

Chord length =

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

### Example Question #5 : Area Of A Circle

The circumference of a circle is inches. Find the area of the circle.

Let .

Explanation:

First we need to find the radius of the circle. The circumference of a circle is , where is the radius of the circle.

The area of a circle is where   is the radius of the circle.

### Example Question #41 : Circles

In the above figure, .

What percent of the figure is shaded gray?

Explanation:

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for  in the formula :

The outer gray ring is the region between the largest and second-largest circles, and has area

The inner gray ring is the region between the second-smallest and smallest circles, and has area

The total area of the gray regions is

Since  out of total area  is gray, the percent of the figure that is gray is

.

### Example Question #42 : Circles

In the above figure, .

Give the ratio of the area of the outer ring to that of the inner circle.

12 to 1

7 to 1

16 to 1

9 to 1

7 to 1

Explanation:

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The four concentric circles have radii 1, 2, 3, and 4, respectively, and their areas can be found by substituting each radius for  in the formula

The areas of the largest circle and the second-largest circle are, respectively,

The difference of their areas, which is the area of the outer ring, is

.

The inner circle has area

.

The ratio of these areas is therefore

, or 7 to 1.

### Example Question #43 : Circles

The above figure depicts a dartboard, in which .

A blindfolded man throws a dart at the target. Disregarding any skill factor and assuming he hits the target, what are the odds against his hitting the white inner circle?

16 to 1

8 to 1

15 to 1

7 to 1

15 to 1

Explanation:

For the sake of simplicity, we will assume that ; this reasoning is independent of the actual length.

The inner and outer circles have radii 1 and 4, respectively, and their areas can be found by substituting each radius for  in the formula :

- this is the white inner circle.

The area of the portion of the target outside the white inner circle is , so the odds against hitting the inner circle are

- that is, 15 to 1 odds against.

### Example Question #21 : Area And Circumference Of A Circle

What is the area of a circle with a diameter of , rounded to the nearest whole number?

Explanation:

The formula for the area of a circle is

Find the radius by dividing 9 by 2:

So the formula for area would now be:

### Example Question #44 : Circles

A circle has a radius of 5 miles, what is its area?

Explanation:

A circle has a radius of 5 miles, what is its area?

Find the area of a circle with the following formula:

We know that r is 5, so we can find our answer with the following:

### Example Question #45 : Circles

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the area of the stamping surface?

Explanation:

While mailing some very important letters, you decide to use your circular rubber stamp. If the stamp has a radius of 3.5 cm, what is the area of the stamping surface?

Find the area of a circle by using the following formula:

Our radius is 3.5 cm, so plug that in:

### Example Question #46 : Circles

You have a circular window in your vacation room. It has a radius of 9 inches. What is the area of the window?

Explanation:

You have a circular window in your vacation room. It has a radius of 9 inches. What is the area of the window?

To find the area of a circle, use the following formula:

Now, we know the radius, so we just need to plug it in and solve.