# High School Math : Circles

## Example Questions

### Example Question #61 : Circles

An original circle has an area of . If the radius is increased by a factor of 3, what is the ratio of the new area to the old area?

Explanation:

The formula for the area of a circle is . If we increase  by a factor of 3, we will increase the area by a factor of 9.

### Example Question #62 : Circles

A square has an area of .  If the side of the the square is the same as the diameter of a circle, what is the area of the circle?

Explanation:

The area of a square is given by so we know that the side of the square is 6 in.  If a circle has a diameter of 6 in, then the radius is 3 in.  So the area of the circle is   or .

### Example Question #21 : Radius

Mary has a decorative plate with a diameter of ten inches. She places the plate on a rectangular placemat with a length of 18 inches and a width of 12 inches. How much of the placemat is visible?

Explanation:

First we will calculate the total area of the placemat:

Next we will calculate the area of the circular place

And

So

We will subtract the area of the plate from the total area

### Example Question #21 : How To Find The Area Of A Circle

The picture above contains both a circle with diameter 4, and a rectangle with length 8 and width 5. Find the area of the shaded region. Round your answer to the nearest integer

Explanation:

First, recall that the diameter of a circle is twice the value of the radius. Therefore a circle with diameter 4 has a radius of 2. Next recall that the area of a circle with radius  is:

The area of the rectangle is the length times the width:

The area of the shaded region is the difference between the 2 areas:

The nearest integer is 27.

### Example Question #21 : Plane Geometry

Allen was running around the park when he lost his keys.  He was running around aimlessly for the past 30 minutes.  When he checked 10 minutes ago, he still had his keys.  Allen guesses that he has been running at about 3m/s.

If Allen can check 1 square kilometer per hour, what is the longest it will take him to find his keys?

Explanation:

Allen has been running for 10 minutes since he lost his keys at 3m/s.  This gives us a maximum distance of  from his current location.  If we move 1800m in all directions, this gives us a circle with radius of 1800m.  The area of this circle is

Our answer, however, is asked for in kilometers.  1800m=1.8km, so our actual area will be  square kilometers.  Since he can search 1 per hour, it will take him at most 10.2 hours to find his keys.

### Example Question #61 : Circles

To the nearest tenth, give the diameter of a circle with area 100 square inches.

Explanation:

The relationship between the radius and the area of a circle can be given as

.

We can substitute  and solve for :

Double this to get the diameter: , which we round to 11.3.

### Example Question #62 : Circles

To the nearest tenth, give the area of a circle with diameter   inches.

Explanation:

The radius of a circle with diameter  inches is half that, or  inches. The area of the circle is

### Example Question #61 : Circles

To the nearest tenth, give the area of a circle with diameter 17 inches.

Explanation:

The radius of a circle with diameter 17 inches is half that, or 8.5 inches. The area of the circle is

### Example Question #61 : Circles

A circle has a radius of . A second circle has a radius of . What is the ratio of the larger circle's area to the smaller circle's area?

Explanation:

The area of a circle is given by the equation , where  is the area and  is the radius of the circle. Use this formula to determine the areas of the two circles:

and

units squared and  units squared.

The ratio of the larger circle to the smaller circle is . Divide each side of the ratio by  to express it in its simplest form, .

### Example Question #33 : Radius

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