<img src="https://d5nxst8fruw4z.cloudfront.net/atrk.gif?account=iA1Pi1a8Dy00ym" style="display:none" height="1" width="1" alt="" />
Dismiss
Skip Navigation

Circumference

Pi times the diameter or the distance around the circle.

Atoms Practice
Estimated7 minsto complete
%
Progress
Practice Circumference
 
 
 
MEMORY METER
This indicates how strong in your memory this concept is
Practice
Progress
Estimated7 minsto complete
%
Practice Now
Turn In
Area and Circumference of Circles
You can use regular polygons with an increasing number of sides to help explain why a circle of radius 1 unit has an area of πr2. Where does the r2 come from in the formula for the area of a circle?

Area and Circumference of a Circle

A circle is a set of points equidistant from a given point. The radius of a circle, r, is the distance from the center of the circle to the circle. All circles are geometrically similar.

A regular polygon is a closed figure that is both equilateral and equiangular. As the number of sides of a regular polygon increases, the polygon looks more and more like a circle.

Previously you have learned that the area of a circle with radius r is given by πr2 and the circumference of a circle with radius r is given by 2πr. In the examples and guided practice, you will derive these formulas by looking at the area and perimeter of regular polygons.

 

Finding the Area  

1. Find the area of a regular octagon inscribed in a circle with radius 1 unit.

Break the octagon into 8 congruent triangles. You will find the area of one triangle and multiply that by 8 to find the area of the whole octagon. Draw in the height for one of the triangles. The angle formed by the height and the radius of the circle is 36016=22.5. Remember that 360 is the number of degrees in a full circle.

Now, you can use trigonometry to find the height and base of the triangle.

  • sin22.5=0.5b1b=2sin22.5b0.7654
  • cos22.5=h1h=cos22.5h0.9239

The area of the triangle is:

A=bh2=(0.7654)(0.9239)20.3536 un2

Therefore, the area of the octagon is:

A=8(0.3536)2.8288 un2

2. Find the area of a regular 60-gon inscribed in a circle with radius 1 unit and the area of a regular 120-gon inscribed in a circle with radius 1 unit.

While you can't accurately draw a regular 60-gon or a regular 180-gon, you can use the method from #1 to find the area of each.

Regular 60-gon: Divide the polygon into 60 congruent triangles. Consider one of those triangles and draw in its height. You will focus on finding the area of this triangle. The angle formed by the height and the radius of the circle is 360120=3.

Use trigonometry to find the height and base of the triangle.

  • sin3=0.5b1b=2sin3b0.1047
  • cos3=h1h=cos3h0.9986

The area of the triangle is:

A=bh2=(0.1047)(0.9986)20.0523 un2

Therefore, the area of the 60-gon is:

A=60(0.0523)3.138 un2

Regular 120-gon: Divide the polygon into 180 congruent triangles. Consider one of those triangles and draw in its height. You will focus on finding the area of this triangle. The angle formed by the height and the radius of the circle is 360360=1.

Use trigonometry to find the height and base of the triangle.

  • sin1=0.5b1b=2sin1b0.0349048
  • cos1=h1h=cos1h0.9998477

The area of the triangle is:

A=bh2=(0.0349048)(0.9998477)20.0174482 un2

Therefore, the area of the 180-gon is:

A=180(0.0174482)3.141 un2

Now, let's explore how area can change based on the number of sides. 

What happens to the area of the regular polygon as the number of sides increases? Make a conjecture about the area of a circle with radius 1 unit.

You have the following information:

Number of Sides

8

60

180

Area

2.8288

3.138

3.141

As the number of sides increases, the regular polygon will get closer and closer to the circle that inscribes it. Therefore, the area of the regular polygon will get closer and closer to the area of the circle. Notice that as the number of sides went from 60 to 180, the area barely changed, staying around 3.14. If you increase the number of sides to 1000, you will find that the area of the regular polygon is still approximately 3.14157. You should recognize these numbers as approximately the value of π.

As the number of sides increases, the polygon becomes closer and closer to a circle, and the area gets closer and closer to π un2. A conjecture would be that the area of a circle with radius 1 unit is π un2.

Examples

Example 1

Earlier, you were asked where the  r2  came from in the formula for the area of a circle.

You can use regular polygons with an increasing number of sides to help explain why a circle of radius 1 unit has an area of π un2. Remember that all circles are similar. To create another circle with radius r from a circle with radius 1 unit, apply a similarity transformation with scale factor k=radius.

The area of the transformed circle is k2 times the area of the original circle. Because the area of the original circle is π and the scale factor is equal to r, the area of the transformed circle is πr2. This is one way to explain why the formula for the area of a circle is πr2.

Example 2

 Find the perimeter of a regular octagon inscribed in a circle with radius 1 unit.

The base of one triangle was b=2sin22.5b0.7654. Therefore, the perimeter of the octagon is P8(0.7654)=6.1232 un.

Example 3

Find the perimeter of a regular 60-gon inscribed in a circle with radius 1 unit.

The base of one triangle was b=2sin3b0.1047. Therefore, the perimeter of the 60-gon is P60(0.1047)=6.282 un.

Example 4

Find the perimeter of a regular 180-gon inscribed in a circle with radius 1 unit.

The base of one triangle was b=2sin1b0.0349048. Therefore, the perimeter of the 180-gon is P180(0.0349048)=6.283 un.

Example 5

Make a conjecture about the circumference of a circle with radius 1 unit.

The perimeters from #2, #3, and #4 are approaching 6.2832π. A conjecture would be that the circumference of a circle with radius 1 unit is 2π un.

Review

Consider a regular n-gon inscribed in a circle with radius 1 unit for questions 1-8.

1. What's the measure of the angle between the radius and the height of one triangle in terms of n?

2. What's the length of the base of one triangle in terms of sine and n?

3. What's the height of one triangle in terms of cosine and n?

4. What's the area of one triangle in terms of sine, cosine, and n?

5. What's the area of the polygon in terms of sine, cosine, and n?

6. What's the perimeter of the polygon in terms of sine, cosine, and n?

7. Let \begin{align*}n=10,000\end{align*}. What is the area of the polygon? What is the perimeter of the polygon? Use your calculator and your answers to #6 and #7.

8. Let \begin{align*}n=1,000,000\end{align*}. What is the area of the polygon? What is the perimeter of the polygon? Are you convinced that the area of a circle with radius 1 unit is \begin{align*}\pi\end{align*} and the circumference of a circle with radius 1 unit is \begin{align*}2 \pi\end{align*}?

9. Explain why the area of a regular polygon inscribed in a circle with radius 1 unit gets closer to \begin{align*}\pi\end{align*} as the number of sides increases.

10. Explain why the perimeter of a regular polygon inscribed in a circle with radius 1 unit gets closer to \begin{align*}2 \pi\end{align*} as the number of sides increases.

11. Use similarity and the fact that the circumference of a circle with radius 1 unit is \begin{align*}2 \pi\end{align*} to explain why the formula for the circumference of a circle with radius \begin{align*}r\end{align*} is \begin{align*}2 \pi r\end{align*}.

12. A circle with radius 3 units is transformed into a circle with radius 5 units. What is the ratio of their areas? What is the ratio of their circumferences?

13. The ratio of the areas of two circles is \begin{align*}25:4\end{align*}. The radius of the smaller circle is 2 units. What's the radius of the larger circle?

14. The ratio of the areas of two circles is \begin{align*}25:9\end{align*}. The radius of the larger circle is 10 units. What's the radius of the smaller circle?

15. The ratio of the areas of two circles is \begin{align*}5:4\end{align*}. The radius of the larger circle is 8 units. What's the circumference of the smaller circle?

Review (Answers)

To see the Review answers, open this PDF file and look for section 8.2. 

Notes/Highlights Having trouble? Report an issue.

Color Highlighted Text Notes
Please to create your own Highlights / Notes
Show More

Vocabulary

Circumference

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

Congruent

Congruent figures are identical in size, shape and measure.

Radius

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

Regular Polygon

A regular polygon is a polygon with all sides the same length and all angles the same measure.

Image Attributions

Explore More

Sign in to explore more, including practice questions and solutions for Circumference.
Please wait...
Please wait...