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Circle Circumference

C = πd; C = 2πr

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Circle Circumference
License: CC BY-NC 3.0

The theme of the school dance this year is “The 50’s.” For decorations, the 7th grade class is gathering old records and gluing a sparkly trim around the edges. How much trim do they need for each record if its diameter is 12 inches?

In this concept, you will learn how to find the circumference of a circle using a given diameter.

Finding the Circumference of a Circle

The perimeter is the total distance around the edges of a two-dimensional shape. To find the perimeter of a shape made from line segments, such as a square or triangle, you simply add the lengths of the sides together.

\begin{align*}P_{\text{Square}}= s+s+s+s \ \text{or} \ P_S=4s\end{align*}  

\begin{align*}P_{\text{Triangle}} = s_1+s_2+s_3\end{align*} 

The perimeter of a circle is called the circumference and is equal to the total distance around the edge.

The diameter, d, of a circle is the straight-line measurement from one point on the circle to another point on the circle that passes directly through the circle’s center.

A circle is a unique shape in that every point along the circumference is exactly the same distance from the circle’s center. This measurement of one half the diameter is called the radius, r.

\begin{align*}d=2r\end{align*} 

License: CC BY-NC 3.0

One of the unique qualities of a circle is that its diameter and circumference have a proportional relationship. This means that no matter what size the circle is, the proportional relationship, or ratio, between its circumference and diameter is always the same.

The formula for the circumference of a circle is the value of pi times the circle’s diameter: \begin{align*}C=\pi d\end{align*} 

Since the diameter is twice the radius, the formula can also be written: \begin{align*}C= 2 \pi r\end{align*} 

You already know that pi is an irrational number and goes on forever. The rounded value of pi, 3.14, is usually used for calculations.

License: CC BY-NC 3.0

Let’s look at an example.

The circles above have different radii and different circumferences. Use these values and the circumference formula to solve for pi.

First, substitute the values of the first circle into the circumference equation.

\begin{align*}6.28= \pi \times 2\end{align*} 

Next, perform the necessary calculations to isolate \begin{align*}\pi\end{align*}.

\begin{align*}\begin{array}{rcl} 6.28 \div 2 &=& \pi \times 2 \div 2 \\ 3.14 &=& \pi \end{array}\end{align*} 

Then, substitute the values of the second circle into the equation.

\begin{align*}12.56= \pi \times 4\end{align*} 

Perform the necessary calculations to isolate \begin{align*}\pi\end{align*}.

\begin{align*}\begin{array}{rcl} 12.56 \div 4 &=& \pi \times 4 \div 4 \\ 3.14 &=& \pi \end{array}\end{align*}

Pi is what is called a constant. It always stays the same.

Examples

Example 1

Earlier, you were given a problem about the school dance decorations.

The students need to know how much sparkly trim they will need to go around each record if its diameter is 12 inches.

First, write down the formula.

\begin{align*}C= \pi d\end{align*} 

Next, substitute in what you know.

\begin{align*}C= 3.14 \times 12\end{align*}

Then, multiply to solve the equation.

\begin{align*}C= 37.68 \ \text{inches}\end{align*}

The answer is the circumference is 21.98 inches. The students will probably round this up to 38 inches.

Example 2

Find the circumference of the circle below.

License: CC BY-NC 3.0

First write down the formula. 


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

Next, substitute in what you know.


\begin{align*}\begin{array}{rcl} C &=& \pi \times 8 \\ C &=& 3.14 \times 8 \end{array}\end{align*}

Then, multiply to solve the equation. 


\begin{align*}C=25.12\end{align*}

If you were to unroll the circle into a flat line, it would be 25.12 inches long. The answer is the circumference, \begin{align*}C = 25.12 \ in\end{align*}.  

Example 3

What is the circumference of the circle below?

License: CC BY-NC 3.0

First, write down the formula.

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

Next, substitute in what you know.

\begin{align*}\begin{array}{rcl} C &=& \pi \times 12.7 \\ C &=& 3.14 \times 12.7 \end{array}\end{align*}

Then, multiply to solve the equation, and round.

\begin{align*}C= 39.878\end{align*}

The answer is the circumference, \begin{align*} C = 39.88 \ m\end{align*}.

Example 4

Find the circumference of the circle given the radius.

License: CC BY-NC 3.0

First, remember that the diameter is twice the radius, and use the formula that includes that variable.

\begin{align*}C= 2 \pi r\end{align*}

Next, substitute in the values that you know.

\begin{align*}C= 2(3.14)(3)\end{align*}

Then, multiply.

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

The answer is 18.84 inches.

Example 5

Find the circumference of a circle with a diameter of 5 inches.

First, write down the formula.

 \begin{align*}C= \pi d\end{align*}

Next, substitute in what you know.

 \begin{align*}C= 3.14 \times 5\end{align*}

Then, multiply to solve the equation.

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

The answer is the circumference, \begin{align*}C = 15.7 \ \text{inches}\end{align*}.

Review

Find the circumference of each circle given the radius or the diameter.

  1. radius = 2 in
  2. diameter = 4 ft
  3. radius = 4.5 in
  4. diameter = 8 meters
  5. radius = 12 inches
  6. diameter = 12 mm
  7. radius = 14 mm
  8. diameter = 13 feet
  9. radius = 10 inches
  10. diameter = 7.5 feet
  11. radius = 2.5 inches
  12. radius = 5.6 feet
  13. diameter = 3.75 feet
  14. diameter = 4.5 feet
  15. diameter = 10.75 meters

Review (Answers)

To see the Review answers, open this PDF file and look for section 9.15.

Resources

 

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Vocabulary

\pi

\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.

Circle

A circle is the set of all points at a specific distance from a given point in two dimensions.

Circumference

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

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.

Perimeter

Perimeter is the distance around a two-dimensional figure.

Pi

\pi (Pi) is the ratio of the circumference of a circle to its diameter. It is an irrational number that is approximately equal to 3.14.

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.

Image Attributions

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