# High School Math : Circles

## Example Questions

### Example Question #10 : Radius

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

Explanation:

The formula for area of a circle is . Unfortunately, the problem gives us a diameter instead of a radius. The good news is that the diameter is equal to twice the length of the radius or, mathematically,

Plug in the given diameter to find the radius:

Plug that into our first equation to solve:

### Example Question #41 : Circles

This figure is a circle with a radius of 3 cm.

What is the area of the circle (cm2)?

Explanation:

The area of the circle is  times radius squared ().

### Example Question #6 : Plane Geometry

If a circle has circumference , what is its area?

Explanation:

If the circumference is , then since  we know . We further know that , so

### Example Question #7 : Plane Geometry

If the equation of a circle is (x – 7)+ (y + 1)= 81, what is the area of the circle?

18π

49π

6561π

81π

81π

Explanation:

The equation is already in a circle equation, and the right side of the equation stands for r2 → r= 81 and r = 9

The area of a circle is πr2, so the area of this circle is 81π.

### Example Question #8 : Plane Geometry

Assume π = 3.14

A man would like to put a circular whirlpool in his backyard. He would like the whirlpool to be six feet wide. His backyard is 8 feet long by 7 feet wide. By state regulation, in order to put a whirlpool in a backyard space, the space must be 1.5 times bigger than the pool. Can the man legally install the whirlpool?

No, because the area of the backyard is 30 square feet and therefore the whirlpool is too big to meet the legal requirement.

No, because the area of the backyard is smaller than the area of the whirlpool.

Yes, because the area of the whirlpool is 28.26 square feet and 1.5 times its area would be less than the area of the backyard.

Yes, because the area of the whirlpool is 18.84 square feet and 1.5 times its area would be less than the area of the backyard.

No, because the area of the whirlpool is 42.39 square feet and 1.5 times its area would be greater than the area of the backyard.

Yes, because the area of the whirlpool is 28.26 square feet and 1.5 times its area would be less than the area of the backyard.

Explanation:

If you answered that the whirlpool’s area is 18.84 feet and therefore fits, you are incorrect because 18.84 is the circumference of the whirlpool, not the area.

If you answered that the area of the whirlpool is 56.52 feet, you multiplied the area of the whirlpool by 1.5 and assumed that that was the correct area, not the legal limit.

If you answered that the area of the backyard was smaller than the area of the whirlpool, you did not calculate area correctly.

And if you thought the area of the backyard was 30 feet, you found the perimeter of the backyard, not the area.

The correct answer is that the area of the whirlpool is 28.26 feet and, when multiplied by 1.5 = 42.39, which is smaller than the area of the backyard, which is 56 square feet.

### Example Question #9 : Plane Geometry

There are two identical circles on a plane that overlap. The radius of both circles is 1. The region in which they overlap has an area of π.

What is the total area of the 2 overlapping circles?

2π

1

π

0

2

π

Explanation:

The total area of both circles is π + π = 2π

Since the region overlaps, we cannot count it twice, so we must subtract it.

we get 2π – π = π

### Example Question #10 : Plane Geometry

A square with a side length of 4 inches is inscribed in a circle, as shown below. What is the area of the unshaded region inside of the circle, in square inches?

8π-8

8π - 16

4π-4

8π-4

2π-4

8π - 16

Explanation:

Using the Pythagorean Theorem, the diameter of the circle (also the diagonal of the square) can be found to be 4√2.  Thus, the radius of the circle is half of the diameter, or 2√2.  The area of the circle is then π(2√2)2, which equals 8π.  Next, the area of the square must be subtracted from the entire circle, yielding an area of 8π-16 square inches.

### Example Question #1 : Radius

If a circle has a circumference of 16π, what would its area be if its radius were halved?

16π

64π

16π

Explanation:

The circumference of a circle = πd where d = diameter.  Therefore, this circle’s diameter must equal 16.  Knowing that diameter = 2 times the radius, we can determine that the radius of this circle = 8.  Halving the radius would give us a new radius of 4.  To find the area of this new circle, use the formula A=πr² where r = radius.  Plug in 4 for r.  Area will equal 16π.

### Example Question #61 : Circles

A star is inscribed in a circle with a diameter of 30, given the area of the star is 345, find the area of the shaded region, rounded to one decimal.

341.5

361.5

351.5

356.5

346.5

361.5

Explanation:

The area of the circle is (30/2)2*3.14 (π) = 706.5, since the shaded region is simply the area difference between the circle and the star, it’s 706.5-345 = 361.5

### Example Question #62 : Circles

The diameter of a circle increases by 100 percent. If the original area is 16π, what is the new area of the circle?

54π

50π

160π

49π

64π