# Geometry: perimeter and circumference of geometric figures

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This module is from Fundamentals of Mathematics by Denny Burzynski and Wade Ellis, Jr. This module discusses perimeter and circumference of geometric figures. By the end of the module students should know what a polygon is, know what perimeter is and how to find it, know what the circumference, diameter, and radius of a circle is and how to find each one, know the meaning of the symbol $\pi$ and its approximating value and know what a formula is and four versions of the circumference formula of a circle.

## Section overview

• Polygons
• Perimeter
• The Number $\pi$
• Formulas

## Polygons

We can make use of conversion skills with denominate numbers to make measure­ments of geometric figures such as rectangles, triangles, and circles. To make these measurements we need to be familiar with several definitions.

## Polygon

A polygon is a closed plane (flat) figure whose sides are line segments (portions of straight lines).

## Perimeter

The perimeter of a polygon is the distance around the polygon.

To find the perimeter of a polygon, we simply add up the lengths of all the sides.

## Sample set a

Find the perimeter of each polygon.

$\begin{array}{ccc}\hfill \text{Perimeter}& =& \text{2 cm}+\text{5 cm}+\text{2 cm}+\text{5 cm}\hfill \\ & =& \text{14 cm}\hfill \end{array}$

Our first observation is that three of the dimensions are missing. However, we can determine the missing measurements using the following process. Let A, B, and C represent the missing measurements. Visualize

$A=\text{12m}-\text{2m}=\text{10m}$
$B=\text{9m}+\text{1m}-\text{2m}=\text{8m}$
$C=\text{12m}-\text{1m}=\text{11m}$

## Practice set a

Find the perimeter of each polygon.

20 ft

26.8 m

49.89 mi

## Circumference

The circumference of a circle is the distance around the circle.

## Diameter

A diameter of a circle is any line segment that passes through the center of the circle and has its endpoints on the circle.

A radius of a circle is any line segment having as its endpoints the center of the circle and a point on the circle.
The radius is one half the diameter.

## The number $\pi$

The symbol $\pi$ , read "pi," represents the nonterminating, nonrepeating decimal number 3.14159 … . This number has been computed to millions of decimal places without the appearance of a repeating block of digits.

For computational purposes, $\pi$ is often approximated as 3.14. We will write $\pi \approx 3\text{.}\text{14}$ to denote that $\pi$ is approximately equal to 3.14. The symbol "≈" means "approximately equal to."

## Formulas

To find the circumference of a circle, we need only know its diameter or radius. We then use a formula for computing the circumference of the circle.

## Formula

A formula is a rule or method for performing a task. In mathematics, a formula is a rule that directs us in computations.

Formulas are usually composed of letters that represent important, but possibly unknown, quantities.

If $C$ , $d$ , and $r$ represent, respectively, the circumference, diameter, and radius of a circle, then the following two formulas give us directions for computing the circum­ference of the circle.

## Circumference formulas

1. $C=\pi d$ or $C\approx \left(3\text{.}\text{14}\right)d$
2. $C=2\pi r$ or $C\approx 2\left(3\text{.}\text{14}\right)r$

## Sample set b

Find the exact circumference of the circle.

Use the formula $C=\pi d$ .

$C=\pi \cdot 7\text{in}\text{.}$

By commutativity of multiplication,

$C=7\text{in}\text{.}\cdot \pi$

$C=7\pi \text{in}\text{.}$ , exactly

This result is exact since $\pi$ has not been approximated.

Find the approximate circumference of the circle.

Use the formula $C=\pi d$ .

$C\approx \left(3\text{.}\text{14}\right)\left(6\text{.}2\right)$

This result is approximate since $\pi$ has been approximated by 3.14.

Find the approximate circumference of a circle with radius 18 inches.

Since we're given that the radius, $r$ , is 18 in., we'll use the formula $C=2\pi r$ .

Find the approximate perimeter of the figure.

We notice that we have two semicircles (half circles).

The larger radius is 6.2 cm.

The smaller radius is $6\text{.}\text{2 cm - 2}\text{.}\text{0 cm}=\text{4}\text{.}\text{2 cm}$ .

The width of the bottom part of the rectangle is 2.0 cm.

## Practice set b

Find the exact circumference of the circle.

9.1 $\pi$ in.

Find the approximate circumference of the circle.

5.652 mm

Find the approximate circumference of the circle with radius 20.1 m.

126.228 m

Find the approximate outside perimeter of

41.634 mm

## Exercises

Find each perimeter or approxi­mate circumference. Use $\pi =3\text{.}\text{14}$ .

21.8 cm

38.14 inches

0.86 m

87.92 m

16.328 cm

0.0771 cm

120.78 m

21.71 inches

43.7 mm

45.68 cm

## Exercises for review

( [link] ) Find the value of $2\frac{8}{\text{13}}\cdot \sqrt{\text{10}\frac{9}{\text{16}}}$ .

8.5 or $\frac{\text{17}}{2}$ or $8\frac{1}{2}$

( [link] ) Find the value of $\frac{8}{\text{15}}+\frac{7}{\text{10}}+\frac{\text{21}}{\text{60}}$ .

( [link] ) Convert $\frac{7}{8}$ to a decimal.

0.875

( [link] ) What is the name given to a quantity that is used as a comparison to determine the measure of another quantity?

( [link] ) Add 42 min 26 sec to 53 min 40 sec and simplify the result.

1 hour 36 minutes 6 seconds

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