How does the area of a circle relate to the area of a rectangle?

#### Watch This

http://www.youtube.com/watch?v=SIKkWLqt2mQ James Sousa: Determine the Area of a Circle

http://www.youtube.com/watch?v=sHtsnC2Mgnk James Sousa: Determine the Circumference of a Circle

#### Guidance

A circle is the set of all points equidistant from a center point. From the center point to the circle is called the
**
radius
**
. From one side of the circle to the other through the center point is called the
**
diameter
**
.

The perimeter of a circle is called its
**
circumference
**
. The ratio between the circumference and diameter of any circle is
.
, or “pi”, is a Greek letter that stands for an irrational number approximately equal to 3.14. Because
is the ratio between the circumference and the diameter, the circumference of a circle is equal to the diameter times
.

The area of a circle is also related to .

**
Example A
**

Find the area and circumference of a circle with radius 5 inches.

**
Solution:
**

Leaving your answer with a symbol in it is called leaving your answer “in terms of ”. This is often preferable because it is the exact answer. As soon as you approximate a value for your answer is not exact. Keep in mind that is not the units. You should still put the appropriate units on your answers.

**
Example B
**

Find the area and circumference of a circle with diameter 16 cm.

**
Solution:
**
If the diameter is 16 cm, then the radius is 8 cm.

**
Example C
**

The shape below is a portion of a circle called a sector. This sector is of the circle. Find the area and perimeter of the sector.

**
Solution:
**
Since the sector is
of the circle, its area will be
the area of the circle. The area of the sector will be
.

The perimeter of the sector is the sum of the lengths of the two radiuses and the arc. Each radius is 4
*
in
*
. The arc is
of the circumference of the full circle. The length of the arc is
.

**
Concept Problem Revisited
**

You can understand the formula for the area of a circle by dissecting a circle into wedges and rearranging them to form a shape that is close to a parallelogram. The parallelogram can then by formed into a shape close to a rectangle.

The lengths of the sides of the “parallelogram” are and . If you imagine cutting the wedges smaller and smaller, the parallelogram will look closer and closer to a rectangle with dimensions and . The area of the rectangle and thus the area of the circle will be .

#### Vocabulary

A
**
circle
**
is the set of all points equidistant from a center point.

**
Equidistant
**
means the same distance away from.

**
Circumference
**
is the perimeter of a circle.

is a Greek letter that stands for an irrational number that is approximately equal to 3.14.

The
**
radius
**
of a circle is the distance from the center point to the circle.

The
**
diameter
**
of a circle is the distance from one side of the circle to the other while passing through the center point.

#### Guided Practice

1. Explain why the formula also works for the circumference of a circle.

2. Find the area of a circle with a circumference of .

3. Find the area and circumference of the sector below. The sector is of the circle.

**
Answers:
**

1. The relationship between the radius and the diameter is . If you substitute for in the formula , you will have or .

2. If the circumference is then the diameter is and the radius is therefore . The area is .

3. .

#### Practice

1. Find the area and circumference of a circle with radius 3
*
in
*
.

2. Find the area and circumference of a circle with radius 6
*
in
*
.

3. Find the area and circumference of a circle with diameter 15
*
cm
*
.

4. Find the area and circumference of a circle with diameter 22
*
in
*
.

5. Find the area of a circle with a circumference of .

6. Find the circumference of a circle with area .

7. The sector below is of a circle. Find the area of the sector.

8. Find the perimeter of the sector in #7.

9. The sector below is of a circle. Find the area of the sector.

10. Find the perimeter of the sector in #9.

11. The sector below is of a circle. Find the area of the sector.

12. Find the perimeter of the sector in #11.

13. The sector below is of a circle. Find the area of the sector.

14. Find the perimeter of the sector in #13.

15. Explain the formula where is the area of a circle, is the circumference of the circle, and is the radius of the circle.