6.2.1: Perimeter and Area of Geometric Figures

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Page ID: 87301

Leah Griffith, Veronica Holbrook, Johnny Johnson & Nancy Garcia
Rio Hondo College

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6.2.1 Learning Objectives

Define polygon
Find the perimeter of a polygon
Find the areas of common polygons

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

$Four shapes, each completely closed, with various numbers of straight line segments as sides.$ $a rectangle$ $a hexagon$ $a polygon with seven sides with no clear pattern. Seven line sements connected together to form a closed figure.$

Not polygons

$Four shapes. One three-sided open box. One oval. One oval-shaped object with one flat side, and one nondescript blob.$ $graphics6.png$ $graphics7.png$ $graphics8.png$

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.

Example 1

Find the perimeter of the rectangle.

$A rectangle with short sides of length 2 cm and long sides of length 5 cm.$

Solution

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

Example 2

$A polygon with sides of the following lengths: 9.2cm, 31mm, 4.2mm, 4.3mm, 1.52cm, and 5.4mm.$

Solution

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

Example 3

$A polygon with eight sides. It is not an octagon, but can be visualized as one large rectangle with two smaller rectangles connected to it.$

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

$A polygon with eight sides. It is not an octagon, but can be visualized as one large rectangle with two smaller rectangles connected to it. The height and width are measured and labeled with variables, A, B, and C.$

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

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

Try It Now 1

Find the perimeter of this triangle.

$A three-sided polygon with sides of the following lengths: 3 ft, 8 ft, and 9 ft.$

Answer: 20 ft

Try It Now 2

Find the perimeter of this polygon.

$A polygon with eight sides. It is not an octagon, but can be visualized as one large square with two smaller rectangles connected to it.$

Answer: 46 cm

Try It Now 3

$A four-sided polygon with sides of the following length: 6.1m, 8.6m, 6.3m, and 5.8m.$

Answer: 26.8 m

Try It Now 4

$A seven-sided polygon with sides of the following lengths: 10.07mi, 3.88mi, 4.54mi, 4.92mi, 12.61, 10.76mi, and 3.11mi.$

Answer: 49.89 mi

Quite often it is necessary to multiply one denominate number by another. To do so, we multiply the number parts together and the unit parts together. For example,

$\begin{array} {rcl} {\text{8 in} \cdot \text{8 in}} & = & {8 \cdot 8 \cdot \text{in} \cdot \text{in}} \\ {} & = & {64 \text{ in}^2} \end{array}$

$\begin{array} {rcl} {\text{4 mm} \cdot \text{4 mm} \cdot \text{4 mm}} & = & {4 \cdot 4 \cdot 4 \cdot \text{mm} \cdot \text{mm} \cdot \text{mm}} \\ {} & = & {64 \text{ mm}^3} \end{array}$

Sometimes the product of units has a physical meaning. In this section, we will examine the meaning of the products $\text{(length unit)}^2$ and $\text{(length unit)}^3$

The Meaning and Notation for Area

The product $\text{(length unit)} \cdot \text{(length unit)} = \text{(length unit)}^2$, or, square length unit (sq length unit), can be interpreted physically as the area of a surface.

Area
The area of a surface is the number of square length units contained in the surface.

For example, 3 sq in means that 3 squares, 1 inch on each side, can be placed precisely on some surface. (The squares may have to be cut and rearranged so they match the shape of the surface.)

We will examine the area of the following geometric figures.

$Triangles, a three-sided polygon, have a height, h, measured from bottom to top, and base, b, measured from one end to the other of the bottom side.$ $Rectangles, a four-sided polygon, have a width, w, in this case the vertical side, and a length, l, in this case the horizontal side.$

$Parallelograms, a four-sided polygon with diagonal sides in the same direction have a height, h, measured as the distance from the bottom to top, and a base, b, measured as the width of the horizontal side.$ $Trapezoids, a four-sided polygon with diagonal sides facing leaning into each other, have a height measured as the distance between the two bases. Trapezoids have two bases of differing lengths, base 1, and base 2.$

Area Formulas

We can determine the areas of these geometric figures using the following formulas.

	Figure	Area Formula	Statement
$A triangle.$	Triangle	$A_T = \dfrac{1}{2} \cdot b \cdot h$	Area of a triangle is one half the base times the height.
$A rectangle.$	Rectangle	$A_R = l \cdot w$	Area of a rectangle is the length times the width.
$A parallelogram.$	Parallelogram	$A_P = b \cdot h$	Area of a parallelogram is base times the height.
$A trapezoid.$	Trapezoid	$A_{Trap} = \dfrac{1}{2} \cdot (b_1 + b_2) \cdot h$	Area of a trapezoid is one half the sum of the two bases times the height.

Finding Areas of Some Common Geometric Figures

Example 4

Find the area of the triangle.

$A triangle with height 6 feet and length 20 feet.$

Solution

$\begin{array} {rcl} {A_T} & = & {\dfrac{1}{2} \cdot b \cdot h} \\ {} & = & {\dfrac{1}{2} \cdot 20 \cdot 5 \text{ sq ft}} \\ {} & = & {10 \cdot 6 \text{ sq ft}} \\ {} & = & {60 \text{ sq ft}} \\ {} & = & {60 \text{ ft}^2} \end{array}$

The area of this triangle is 60 sq ft, which is often written as 60 $\text{ft}^2$.

Example 5

Find the area of the rectangle.

$A rectangle with width 4 feet 2 inches and height 8 inches.$

Solution

Let's first convert 4 ft 2 in to inches. Since we wish to convert to inches, we'll use the unit fraction $\dfrac{\text{12 in}}{\text{1 ft}}$ since it has inches in the numerator. Then,

$\begin{array} {rcl} {\text{4 ft}} & = & {\dfrac{\text{4 ft}}{1} \cdot \dfrac{\text{12 in}}{\text{1 ft}}} \\ {} & = & {\dfrac{4 \cancel{\text{ ft}}}{1} \cdot \dfrac{\text{12 in}}{1 \cancel{\text{ ft}}}} \\ {} & = & {\text{48 in}} \end{array}$

Thus, $\text{4 ft 2 in = 48 in + 2 in = 50 in}$

$\begin{array} {rcl} {A_R} & = & {l \cdot w} \\ {} & = & {\text{50 in} \cdot \text{8 in}} \\ {} & = & {400 \text{ sq in}} \end{array}$

The area of this rectangle is 400 sq in.

Example 6

Find the area of the parallelogram.

$A parallelogram with base 10.3cm and height 6.2cm$

Solution

$\begin{array} {rcl} {A_P} & = & {b \cdot h} \\ {} & = & {\text{10.3 cm} \cdot \text{6.2 cm}} \\ {} & = & {63.86 \text{ sq cm}} \end{array}$

The area of this parallelogram is 63.86 sq cm.

Example 7

Find the area of the trapezoid.

$A trapezoid with height 4.1mm, bottom base 20.4mm, and top base 14.5mm.$

Solution

$\begin{array} {rcl} {A_{Trap}} & = & {\dfrac{1}{2} \cdot (b_1 + b_2) \cdot h} \\ {} & = & {\dfrac{1}{2} \cdot (\text{14.5 mm + 20.4 mm}) \cdot (4.1 \text{ mm})} \\ {} & = & {\dfrac{1}{2} \cdot (\text{34.9 mm}) \cdot (4.1 \text{ mm})} \\ {} & = & {\dfrac{1}{2} \cdot \text{(143.09 sq mm)}} \\ {} & = & {71.545 \text{ sq mm}} \end{array}$

The area of this trapezoid is 71.545 sq mm.

Try It Now 5

Find the area of each of the following geometric figures.

$A triangle with base 18cm and height 4cm.$

Answer: 36 sq cm

Try It Now 6

Find the area of the rectangle

$A rectangle with base 9.26mm and height 4.05mm.$

Answer: 37.503 sq mm

Try It Now 7

$A parallelogram with base 5.1in and height 2.6in.$

Answer: 13.26 sq in.

Try It Now 8

$A trapezoid with height 15mi, bottom base 32mi, and top base 17mi.$

Answer: 367.5 sq mi

Composite Figures

Definition: Composite Figures

A composite figure is a figure made up of two or more geometric figures.

When determining the area of a composite figure it is best to determine what shapes the composite figure is made of first. Once you have determined the shapes, find the area for those individual shapes and add them together.

Example 8

Find the area of the composite figure.

$A polygon with eight sides. It is not an octagon, but can be visualized as one large rectangle with two smaller rectangles connected to it.$

Solution

This figure is made up of two squares and one rectangle. There is a square with 1 cm sides and one with 2 cm sides. The rectangle has a 12 cm side, but to figure out the other side we need to subtract 9 cm$-$2 cm $=$7 cm.

$\begin{array} {rcl} {A_{figure}} & = & {l \cdot w + l \cdot w + l \cdot w} \\ {} & = & {1 \text{ cm} \cdot 1 \text{ cm} + 2 \text{ cm} \cdot 2 \text{ cm}+12 \text{ cm} \cdot 7 \text{ cm}} \\ {} & = & {1 \text{ cm}^2+2 \text{ cm}^2+84 \text{ cm}^2} \\ {} & = & {87 \text{sq cm}} \end{array}$

The area of the composite figure is 87 sq. cm.

Try It Now 9

Find the area of the composite figure.

$a ploygon with four sides. The left side is 15 in tall. It connects to the bottom side at a right angle. The bottom is 22 in wide, it connects to the right side at a right angle. The right side is 17 in. The top connects the 15 in side and the 17 in side and forms a diagonal side.$

Answer

This shape is composed of a rectangle and a triangle.

The area of the composite figure is 352 sq in.

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