### Circumference Definition and Formula

The circumference of a circle is its perimeter or distance around it. It is denoted by C in math formulas and has units of distance, such as millimeters (mm), centimeters (cm), meters (m), or inches (in). It is related to the radius, diameter, and pi using the following equations:

C = πd

C = 2πr

Where d is the diameter of the circle, r is its radius, and π is pi. The diameter of a circle is the longest distance across it, which you can measure from any point on the circle, going through its center or origin, to the connecting point on the far side.

The radius is one-half the diameter or it can be measured from the origin of the circle out to its edge.

π (pi) is a mathematical constant that relates a circle's circumference to its diameter. It is an irrational number, so it doesn't have a decimal representation. In calculations, most people use 3.14 or 3.14159. Sometimes it is approximated by the fraction 22/7.

### Find the Circumference - Examples

(1) You measure the diameter of a circle to be 8.5 cm. Find the circumference.

To solve this, simply enter the diameter in the equation. Remember to report your answer with the proper units.

C = πd

C = 3.14 * (8.5 cm)

C = 26.69 cm, which you should round up to 26.7 cm

(2) You want to know the circumference of a pot that has a radius of 4.5 inches.

For this problem, you can either use the formula that includes radius or you can remember the diameter is twice the radius and use that formula. Here's the solution, using the formula with radius:

C = 2πr

C = 2 * 3.14 * (4.5 in)

C = 28.26 inches or 28 inches, if you use the same number of significant figures as your measurement.

(3) You measure a can and find it is 12 inches in circumference. What is its diameter? What is its radius?

Although a can is a cylinder, it still has a circumference because a cylinder is basically a stack of circles. To solve this problem, you need to rearrange the equations:

C = πd may be rewritten as:

C/π = d

Plugging in the circumference value and solving for d:

C/π = d

(12 inches) / π = d

12 / 3.14 = d

3.82 inches = diameter (let's call it 3.8 inches)

You could play the same game to rearrange a formula to solve for the radius, but if you have the diameter already, the easiest way to get the radius is to divide it in half:

radius = 1/2 * diameter

radius = (0.5) *(3.82 inches) [remember, 1/2 = 0.5]

radius = 1.9 inches

### Notes About Estimates and Reporting Your Answer

- You should always check your work. One quick way to estimate whether your circumference answer is reasonable is to check to see if it's a bit more than 3 times larger than the diameter or slightly over 6 times larger than the radius.
- You should match the number of significant figures you use for pi to that of the significance of the other values you are given. If you don't know what significant figures are or aren't asked to work with them, don't worry about this. Basically, this means if you have a very precise distance measurement, like 1244.56 meters (6 significant figures), you want to use 3.14159 for pi and not 3.14. Otherwise, you'll end up reporting a less precise answer.

### Finding the Area of a Circle

If you know the circumference, radius, or diameter of a circle, you can also find its area. Area represents the space enclosed within a circle. It's given in units of distance squared, such as cm^{2} or m^{2}.

The area of a circle is given by the formulas:

A = πr^{2} (Area equals pi times the radius squared.)

A = π(1/2 d)^{2} (Area equals pi times one-half the diameter squared.)

A = π(C/2π)^{2} (Area equals pi times the square of the circumference divided by two times pi.)