# 9.4: Perimeter and Circumference of Geometric Figures

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$

( \newcommand{\kernel}{\mathrm{null}\,}\) $$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\id}{\mathrm{id}}$$

$$\newcommand{\Span}{\mathrm{span}}$$

$$\newcommand{\kernel}{\mathrm{null}\,}$$

$$\newcommand{\range}{\mathrm{range}\,}$$

$$\newcommand{\RealPart}{\mathrm{Re}}$$

$$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$

$$\newcommand{\Argument}{\mathrm{Arg}}$$

$$\newcommand{\norm}[1]{\| #1 \|}$$

$$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$

$$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\AA}{\unicode[.8,0]{x212B}}$$

$$\newcommand{\vectorA}[1]{\vec{#1}} % arrow$$

$$\newcommand{\vectorAt}[1]{\vec{\text{#1}}} % arrow$$

$$\newcommand{\vectorB}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vectorC}[1]{\textbf{#1}}$$

$$\newcommand{\vectorD}[1]{\overrightarrow{#1}}$$

$$\newcommand{\vectorDt}[1]{\overrightarrow{\text{#1}}}$$

$$\newcommand{\vectE}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash{\mathbf {#1}}}}$$

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$

$$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$

$$\newcommand{\avec}{\mathbf a}$$ $$\newcommand{\bvec}{\mathbf b}$$ $$\newcommand{\cvec}{\mathbf c}$$ $$\newcommand{\dvec}{\mathbf d}$$ $$\newcommand{\dtil}{\widetilde{\mathbf d}}$$ $$\newcommand{\evec}{\mathbf e}$$ $$\newcommand{\fvec}{\mathbf f}$$ $$\newcommand{\nvec}{\mathbf n}$$ $$\newcommand{\pvec}{\mathbf p}$$ $$\newcommand{\qvec}{\mathbf q}$$ $$\newcommand{\svec}{\mathbf s}$$ $$\newcommand{\tvec}{\mathbf t}$$ $$\newcommand{\uvec}{\mathbf u}$$ $$\newcommand{\vvec}{\mathbf v}$$ $$\newcommand{\wvec}{\mathbf w}$$ $$\newcommand{\xvec}{\mathbf x}$$ $$\newcommand{\yvec}{\mathbf y}$$ $$\newcommand{\zvec}{\mathbf z}$$ $$\newcommand{\rvec}{\mathbf r}$$ $$\newcommand{\mvec}{\mathbf m}$$ $$\newcommand{\zerovec}{\mathbf 0}$$ $$\newcommand{\onevec}{\mathbf 1}$$ $$\newcommand{\real}{\mathbb R}$$ $$\newcommand{\twovec}[2]{\left[\begin{array}{r}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\ctwovec}[2]{\left[\begin{array}{c}#1 \\ #2 \end{array}\right]}$$ $$\newcommand{\threevec}[3]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\cthreevec}[3]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \end{array}\right]}$$ $$\newcommand{\fourvec}[4]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\cfourvec}[4]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \end{array}\right]}$$ $$\newcommand{\fivevec}[5]{\left[\begin{array}{r}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\cfivevec}[5]{\left[\begin{array}{c}#1 \\ #2 \\ #3 \\ #4 \\ #5 \\ \end{array}\right]}$$ $$\newcommand{\mattwo}[4]{\left[\begin{array}{rr}#1 \amp #2 \\ #3 \amp #4 \\ \end{array}\right]}$$ $$\newcommand{\laspan}[1]{\text{Span}\{#1\}}$$ $$\newcommand{\bcal}{\cal B}$$ $$\newcommand{\ccal}{\cal C}$$ $$\newcommand{\scal}{\cal S}$$ $$\newcommand{\wcal}{\cal W}$$ $$\newcommand{\ecal}{\cal E}$$ $$\newcommand{\coords}[2]{\left\{#1\right\}_{#2}}$$ $$\newcommand{\gray}[1]{\color{gray}{#1}}$$ $$\newcommand{\lgray}[1]{\color{lightgray}{#1}}$$ $$\newcommand{\rank}{\operatorname{rank}}$$ $$\newcommand{\row}{\text{Row}}$$ $$\newcommand{\col}{\text{Col}}$$ $$\renewcommand{\row}{\text{Row}}$$ $$\newcommand{\nul}{\text{Nul}}$$ $$\newcommand{\var}{\text{Var}}$$ $$\newcommand{\corr}{\text{corr}}$$ $$\newcommand{\len}[1]{\left|#1\right|}$$ $$\newcommand{\bbar}{\overline{\bvec}}$$ $$\newcommand{\bhat}{\widehat{\bvec}}$$ $$\newcommand{\bperp}{\bvec^\perp}$$ $$\newcommand{\xhat}{\widehat{\xvec}}$$ $$\newcommand{\vhat}{\widehat{\vvec}}$$ $$\newcommand{\uhat}{\widehat{\uvec}}$$ $$\newcommand{\what}{\widehat{\wvec}}$$ $$\newcommand{\Sighat}{\widehat{\Sigma}}$$ $$\newcommand{\lt}{<}$$ $$\newcommand{\gt}{>}$$ $$\newcommand{\amp}{&}$$ $$\definecolor{fillinmathshade}{gray}{0.9}$$
##### Learning Objectives
• 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 ππ and its approximating value
• know what a formula is and four versions of the circumference formula of a circle

## 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.

##### Definition: Polygon

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

Polygons

Not polygons

## Perimeter

##### Definition: 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.

Solution

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

##### Sample Set A

Solution

$$\begin{array} {rcr} {\text{Perimeter}} & = & {\text{3.1 mm}} \\ {} & & {\text{4.2 mm}} \\ {} & & {\text{4.3 mm}} \\ {} & & {\text{1.52 mm}} \\ {} & & {\text{5.4 mm}} \\ {} & & {\underline{\text{+ 9.2 mm}}} \\ {} & & {\text{27.72 mm}} \end{array}$$

##### Sample Set A

Solution

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

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

$$\begin{array} {rcr} {\text{Perimeter}} & = & {\text{8 m}} \\ {} & & {\text{10 m}} \\ {} & & {\text{2 m}} \\ {} & & {\text{2 m}} \\ {} & & {\text{9 m}} \\ {} & & {\text{11 m}} \\ {} & & {\text{1 m}} \\ {} & & {\underline{\text{+ 1 m}}} \\ {} & & {\text{44 m}} \end{array}$$

Practice Set A

Find the perimeter of each polygon.

20 ft

Practice Set A

26.8 m

Practice Set A

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 $$\pi \approx 3.14$$ 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 $$C \approx (3.14) d$$
2. $$C = 2 \pi r$$ or $$C \approx 2 (3.14) r$$
##### Sample Set B

Find the exact circumference of the circle.

Solution

Use the formula $$C = \pi d$$.

$$C = \pi \cdot 7\ in.$$

By commutativity of multiplication,

$$C = 7\ in. \cdot \pi$$

$$C = 7 \pi in.$$, exactly

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

##### Sample Set B

Find the approximate circumference of the circle.

Solution

Use the formula $$C = \pi d$$.

$$C \approx (3.14)(6.2)$$

$$C \approx 19.648 \text{ mm}$$

This result is approximate since ππ has been approximated by 3.14.

##### Sample Set B

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

Solution

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

$$C \approx (2)(3.14)(18 \text{ in.})$$

$$C \approx 113.04 \text{ in.}$$

##### Sample Set B

Find the approximate perimeter of the figure.

Solution

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

The larger radius is 6.2 cm.

The smaller radius is $$\text{6.2 cm - 2.0 cm = 4.2 cm.}$$

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

$$\begin{array} {rcll} {\text{Perimeter}} & = & {\text{2.0 cm}} & {} \\ {} & & {\text{5.1 cm}} & {} \\ {} & & {\text{2.0 cm}} & {} \\ {} & & {\text{5.1 cm}} & {} \\ {} & & {(0.5) \cdot (2) \cdot (3.14) \cdot \text{(6.2 com)}} & {\text{Circumference of outer semicircle.}} \\ {} & \ \ + & {\underline{(0.5) \cdot (2) \cdot (3.14) \cdot \text{(4.2 com)}}} & {\text{Circumference of inner semicircle.}} \\ {} & & {} & {\text{6.2 cm - 2.0 cm = 4.2 cm}} \\ {} & & {} & {\text{The 0.5 appears because we want the}} \\ {} & & {} & {\text{perimeter of only half a circle.}} \end{array}$$

$$\begin{array} {rcr} {\text{Perimeter}} & \approx & {\text{2.0 cm}} \\ {} & & {\text{5.1 cm}} \\ {} & & {\text{2.0 cm}} \\ {} & & {\text{5.1 cm}} \\ {} & & {\text{19.468 cm}} \\ {} & & {\underline{\text{+13.188 cm}}} \\ {} & & {\text{48.856 cm}} \end{array}$$

Practice Set B

Find the exact circumference of the circle.

$$9.1 \pi$$ in.

Practice Set B

Find the approximate circumference of the circle.

5.652 mm

Practice Set B

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

126.228 m

Practice Set B

Find the approximate outside perimeter of

41.634 mm

## Exercises

Find each perimeter or approxi­mate circumference. Use $$\pi = 3.14$$.

Exercise $$\PageIndex{1}$$

21.8 cm

Exercise $$\PageIndex{2}$$

Exercise $$\PageIndex{3}$$

38.14 inches

Exercise $$\PageIndex{4}$$

Exercise $$\PageIndex{5}$$

0.86 m

Exercise $$\PageIndex{6}$$

Exercise $$\PageIndex{7}$$

87.92 m

Exercise $$\PageIndex{8}$$

Exercise $$\PageIndex{9}$$

16.328 cm

Exercise $$\PageIndex{10}$$

Exercise $$\PageIndex{11}$$

0.0771 cm

Exercise $$\PageIndex{12}$$

Exercise $$\PageIndex{13}$$

120.78 m

Exercise $$\PageIndex{14}$$

Exercise $$\PageIndex{15}$$

21.71 inches

Exercise $$\PageIndex{16}$$

Exercise $$\PageIndex{17}$$

43.7 mm

Exercise $$\PageIndex{18}$$

Exercise $$\PageIndex{19}$$

45.68 cm

Exercise $$\PageIndex{20}$$

#### Exercises for Review

Exercise $$\PageIndex{19}$$

Find the value of $$2 \dfrac{8}{13} \cdot \sqrt{10 \dfrac{9}{16}}$$.

8.5 or $$\dfrac{17}{2}$$ or $$8 \dfrac{1}{2}$$

Exercise $$\PageIndex{20}$$

Find the value of $$\dfrac{8}{15} + \dfrac{7}{10} + \dfrac{21}{60}$$.

Exercise $$\PageIndex{19}$$

Convert $$\dfrac{7}{8}$$ to a decimal.

0.875

Exercise $$\PageIndex{20}$$

What is the name given to a quantity that is used as a comparison to determine the measure of another quantity?

Exercise $$\PageIndex{19}$$

Add 42 min 26 sec to 53 min 40 sec and simplify the result.