A circle is a shape with all points the same distance from the center. It is named by the center. The circle to the left is called circle A since the center is at point A. If you measure the distance around a circle and divide it by the distance across the circle through the center, you will always come close to a particular value, depending upon the accuracy of your measurement. This value is approximately 3.14159265358979323846... We use the Greek letter (pronounced Pi) to represent this value. The number goes on forever. However, using computers, has been calculated to over 1 trillion digits past the decimal point.
The distance around a circle is called the circumference. The distance across a circle through the center is called the diameter. is the ratio of the circumference of a circle to the diameter. Thus, for any circle, if you divide the circumference by the diameter, you get a value close to . This relationship is expressed in the following formula:
where is circumference and is diameter. You can test this formula at home with a round dinner plate. If you measure the circumference and the diameter of the plate and then divide by , your quotient should come close to . Another way to write this formula is: where · means multiply. This second formula is commonly used in problems where the diameter is given and the circumference is not known (see the examples below).
The radius of a circle is the distance from the center of a circle to any point on the circle. If you place two radii end-to-end in a circle, you would have the same length as one diameter. Thus, the diameter of a circle is twice as long as the radius. This relationship is expressed in the following formula: , where is the diameter and is the radius.
Circumference, diameter and radii are measured in linear units, such as inches and centimeters. A circle has many different radii and many different diameters, each passing through the center. A real-life example of a radius is the spoke of a bicycle wheel. A 9-inch pizza is an example of a diameter: when one makes the first cut to slice a round pizza pie in half, this cut is the diameter of the pizza. So a 9-inch pizza has a 9-inch diameter. Let's look at some examples of finding the circumference of a circle. In these examples, we will use = 3.14 to simplify our calculations.
Example 1: The radius of a circle is 2 inches. What is the diameter?
Solution:
= 2 · (2 in)
= 4 in
Example 2: The diameter of a circle is 3 centimeters. What is the circumference?
Solution:
= *
= 3.14 · (3 cm)
= 9.42 cm
Example 3: The radius of a circle is 2 inches. What is the circumference?
Solution:
= 2 · (2 in)
= 4 in
= *
= 3.14 · (4 in)
= 12.56 in
Example 4: The circumference of a circle is 15.7 centimeters. What is the diameter?
Solution:
= *
15.7 cm = 3.14 ·
15.7 cm ÷ 3.14 =
= 15.7 cm ÷ 3.14
= 5 cm
Summary: The number is the ratio of the circumference of a circle to its diameter. The value of is approximately 3.14159265358979323846...The diameter of a circle is twice the radius. Given the diameter or radius of a circle, we can find the circumference. We can also find the diameter (and radius) of a circle given the circumference. The formulas for diameter and circumference of a circle are listed below. We round to 3.14 in order to simplify our calculations.
= *
Exercises
Directions: Read each question below. Click once in an ANSWER BOX and type in your answer; then click ENTER. After you click ENTER, a message will appear in the RESULTS BOX to indicate whether your answer is correct or incorrect. To start over, click CLEAR. Use = 3.14 to calculate your answers.
1. | The diameter of a nickel is 2 centimeters. What is the circumference? |
2. | The circumference of a bicycle wheel is 50.24 inches. What is the diameter? |
3. | The radius of a circular rug is 4 feet. What is the circumference? |
4. | The circumference of a compact disc is 28.26 centimeters. What is the radius? |
5. | The diameter of your bicycle wheel is 25 inches. How far will you move in one turn of your wheel? |