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Circle Circumference

C = πd; C = 2πr

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Circle Circumference
Credit: Chris Graziolli
Source: https://www.flickr.com/photos/105262285@N05/10253010234/in/photolist-gC2mJw-gC2nxA-gC2UF9-gC2ocm-gC2PNy-noZPQw-ptRdoK-nwR3UA-9tUm3H-5FdgBb-6QJrA-rBPVt8-aJfUD2-H6MA8-2jaK5-8rX3tY-ohRG7R-8rX44q-bA2jbR-a5FMNo-cYB5Nj-8rTXXv-ebYEGa-bboGxn-8RdEaX-nfk5oM-H6MA4-nfjW9x-nwPCCn-dzr45Z-6B7Ka4-nwyFFd-nwwQfa-gZVYN-p97DEC-nwDtvd-nyoNHD-nf8sau-8s8vuk-8RdCBc-nqKY35-5kRRU5-aPsLqR-a7vkzv-umN4qp-nfkkNL-nuM4rE-bAbjWd-bA2kjP-bA2jdF
License: CC BY-NC 3.0

Tyrone has been hired to build a deck around a circular above-ground pool. Before he starts the project, he needs to know the circumference of the pool. Tyrone doesn’t have a tape measure long enough to go completely around the pool, but he knows that the pool is 30 feet across. How can Tyrone use the information that he already knows to determine the circumference of the pool?

In this concept, you will learn to find the circumference of circles given the radius or diameter.

Finding Circumference

The circumference of a circle is the distance around the perimeter or edge of the circle. The distance across the middle of a circle is its diameter, and the distance from any point on the perimeter to the center is its radius.

To figure out the circumference of the circle, multiply the diameter of the circle and pi or 3.14.

\begin{align*}C = d \pi\end{align*}

Remember, whenever you see the symbol for pi, \begin{align*}\pi\end{align*}, to substitute 3.14 in when multiplying.

Let’s look at an example.

Find the circumference of the circle.

License: CC BY-NC 3.0

The diameter of the circle is 6 inches. Substitute this given information into the circumference formula and solve.

\begin{align*}\begin{array}{rcl} C &=& d \pi \\ C &= & 6(3.14) \\ C &=& 18.84 \ \text{inches} \end{array}\end{align*}

If you were given the radius of a circle instead of the diameter, you could still use this information to figure out the circumference.

Here’s an example.

Find the circumference of the circle.

License: CC BY-NC 3.0

You can solve this problem in two ways:

  1. The radius is half the diameter, you could double the radius and then use the formula for diameter to find the circumference.
  2. Use this formula: \begin{align*}C = 2 \pi r\end{align*}

Let’s use the formula to find the circumference of the circle.

\begin{align*}\begin{array}{rcl} C &=& 2(3.14)(4) \\ C &=& 3.14(8) \\ C &=& 25.12 \ cm \end{array}\end{align*}

The answer is 25.12 cm.

Examples

Example 1

Earlier, you were given a problem about Tyrone and his attempt to measure the circumference of the pool without a tape measure.

The pool’s diameter is 30 feet. The only other thing Tyrone needs to find the circumference of the pool is the equation relating diameter and circumference.

\begin{align*}C = d \pi \end{align*}

To find the circumference of the pool, first, substitute the given information into the equation.

\begin{align*}C = 30 \pi\end{align*}

Next, substitute 3.14 for pi.

\begin{align*}C = 30(3.14)\end{align*}

Then, multiply.

\begin{align*}C = 94.2 \ \text{feet}\end{align*}

The pool’s circumference is 94.2 feet.

Example 2

Find the circumference of a circle using the given information.

A circle with a diameter of 5.5 feet

First, substitute the given information into the equation.

\begin{align*}\begin{array}{rcl} C &=& d \pi \\ C &=& 5.5(3.14) \end{array}\end{align*}

Next, multiply.

\begin{align*}C = 17.27 \ \text{feet}\end{align*}

The answer is 17.27 feet.

Example 3

Find the circumference given the diameter.

\begin{align*}d = 5 \ in\end{align*} 

First, substitute the given information into the equation.

\begin{align*}\begin{array}{rcl} C &=& d \pi \\ C &=& 5(3.14) \end{array}\end{align*}

Next, multiply.

\begin{align*}C = 15.7 \ \text{inches}\end{align*}

The answer is 15.7 inches.

Example 4

Find the circumference given the radius.

\begin{align*}r = 3 \ in\end{align*}

First, substitute the given information into the equation.

\begin{align*}\begin{array}{rcl} C &=& 2r \pi \\ C &=& 2(3)(3.14) \end{array}\end{align*}

Next, multiply.

\begin{align*}C = 18.84 \ \text{inches}\end{align*}

The answer is 18.84 inches.

Example 5

Find the circumference given the diameter.

\begin{align*}d = 2.5 \ cm\end{align*}

First, substitute the given information into the equation.

\begin{align*}\begin{array}{rcl} C &=& d \pi \\ C &=& 2.5(3.14) \end{array}\end{align*}

Next, multiply.

\begin{align*}C = 7.85 \ cm\end{align*}

The answer is 7.85 cm.

Review

Find the circumference of each circle with the given diameter.

  1. \begin{align*}d = 5 \ in.\end{align*} 
  2. \begin{align*}d = 8 \ in.\end{align*} 
  3. \begin{align*}d = 9 \ cm.\end{align*} 
  4. \begin{align*}d = 3 \ cm.\end{align*} 
  5. \begin{align*}d = 10 \ ft.\end{align*} 
  6. \begin{align*}d = 15 \ ft.\end{align*} 
  7. \begin{align*}d = 11 \ m.\end{align*} 
  8. \begin{align*}d = 13 \ ft.\end{align*} 
  9. \begin{align*}d = 17 \ ft.\end{align*} 
  10. \begin{align*}d = 20 \ in.\end{align*} 

Find the circumference of each circle with the given radius.

  1. \begin{align*} r = 2.5 \ in\end{align*} 
  2. \begin{align*} r = 4 \ in\end{align*} 
  3. \begin{align*} r = 4.5 \ cm\end{align*} 
  4. \begin{align*} r = 1.5 \ cm\end{align*} 
  5. \begin{align*} r = 5 \ ft\end{align*} 
  6. \begin{align*} r = 7.5 \ ft\end{align*} 
  7. \begin{align*} r = 5.5 \ m\end{align*} 
  8. \begin{align*} r = 6.5 \ ft.\end{align*} 
  9. \begin{align*} r = 8.5 \ ft\end{align*} 
  10. \begin{align*}r = 10 \ in.\end{align*} 

Review (Answers)

To see the Review answers, open this PDF file and look for section 10.6. 

Resources

 

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Vocabulary

Circumference

The circumference of a circle is the measure of the distance around the outside edge of a circle.

Diameter

Diameter is the measure of the distance across the center of a circle. The diameter is equal to twice the measure of the radius.

Hypotenuse

The hypotenuse of a right triangle is the longest side of the right triangle. It is across from the right angle.

Legs of a Right Triangle

The legs of a right triangle are the two shorter sides of the right triangle. Legs are adjacent to the right angle.

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.

Radius

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

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