Geometry: perimeter and circumference of geometric figures

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

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

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

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

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

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 $C=\pi d$ .

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

$C\approx \text{19}\text{.}\text{648}\text{ mm}$

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 $C=2\pi r$ .

$C\approx \left(2\right)\left(3\text{.}\text{14}\right)\left(\text{18}\text{ in}\text{.}\right)$

$C\approx \text{113}\text{.}\text{04}\text{ in}\text{.}$

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.

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

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