# ISEE Upper Level Math : How to find the area of a circle

## Example Questions

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### Example Question #1 : Area Of A Circle

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 #7 : 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:

Then, solve for area:

### Example Question #1 : 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 : 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 in terms of .

Explanation:

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

,

where   is the diameter of the circle and is approximately .

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

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 #1 : How To Find The Area Of A Circle

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 #1 : 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 #2 : How To Find The Area Of A Circle

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 #1 : How To Find The Area Of A Circle

In the above figure, .

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

7 to 1

9 to 1

16 to 1

12 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 #41 : 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

15 to 1

8 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.

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