Area of Polygons

By Sara Gregurash

AreaThe area of a polygon measures the

size of the region that the figure occupies. It is 2-dimensional like a table cloth. It is expressed in terms of some square unit. Some examples of the units used are square meters, square centimeters, square inches, or square kilometers.

Area of a triangle To find the area of a triangle, you have to

multiply the base by its height and divide by 2. You have to divide by two because a parallelogram can be divided into 2 triangles. The area of each triangle (2 of them) of a parallelogram is equal to one-half the area of the parallelogram. The equation is:

A = ½ (bh) The b stands for the base of a triangle.

The h stands for height. These are shown in the two diagrams below.

Base and Height The base and height of a triangle have to be

perpendicular. The base is a side of the triangle, but the height isn’t always a side of the triangle. When you have a right triangle, then the height is the side of the triangle because it is perpendicular to the base. The sides of a triangle aren’t always perpendicular to the base. For example, when you have an obtuse triangle the side is slanted (this is shown below).

b = 4 cm

h = 8 cm

Area of a right triangle Let’s practice finding the area of triangles. The

same formula is used to find the area of all triangles. I will show you how to find the area of right, acute, obtuse, scalene, and equilateral triangles. Here is an example using right triangles.

For this triangle, the base is 6 cm and the height is 9 cm so you plug those numbers into the equation. It looks like this:

A = ½ (2 in · 4 in) A = ½ (8 in2) A = 4 in2

b = 2 in

h = 4 in

Area of an acute and obtuse triangle

The base of this acute triangle is 16 centimeters and the height is 7 centimeters.

A = ½ (16 cm · 7 cm) A = ½ (112 cm2) A = 56 cm2

The base of this obtuse triangle is 4 inches and the height is 8 inches.

A = ½ (8 in · 4 in) A = ½ (32 in2) A = 16 in2

b = 16 cm

b = 7 cm

h = 8 cm

b = 4 cm

Area of a scalene triangle

A scalene triangle has all sides of different lengths.

The base of this scalene triangle is 80 millimeters and the height is 22 millimeters.

A = ½ (b · h)A = ½ (80 mm · 22 mm)A = ½ (1760 mm 2 )A = 880 mm 2 b = 80 mm

h = 22 mm

s =50 mm

Area of an equilateral triangle

An equilateral triangle is a triangle with three equal length sides and the angles are all equal.

To find the area of the equilateral triangle below with a side of 21 centimeters and the altitude is 14 centimeters, you have to use the formula.

A = ½ base or side · altitude (height)A = ½ (21 cm · 14 cm) A = ½ (294 cm 2 )A = 147 cm 2 a = 14 cm

side (base) = 21 cm

Math Goodies Lesson is a good site to view to practice your skills on area of

triangles.

Parallelograms A parallelogram is a four-sided shape formed by

two pairs of parallel lines. The opposite sides have the same length and the opposite angles have the same degree (size).

To find the area of a parallelogram, you have to multiply its base by its height. The formula looks like this:

A = b · h (b is the base and h is the height) The base and height of a parallelogram must be

perpendicular. The lateral sides of a parallelogram are not perpendicular to the base, so a dotted line is drawn to represent the height.

h

b

Area of a parallelogram To find the area of this parallelogram, you

multiply the base of 8 centimeters and the height of 3 centimeters.

A = b · hA = 8 cm · 3 cmA = 24 cm2

h = 3 cm

b = 8 cm

This site has a lesson about the area of parallelograms.

Trapezoids A trapezoid is a four-sided figure with one pair of

parallel sides. The bases of the trapezoid below are parallel. To find the area of a trapezoid, you have to multiply the sum of its bases by the height and divided by 2. The formula looks like this:

A = 1/2 · (b1 + b2) · h ( b1 is base1, b2 is base2, and h is the height)

Each base of a trapezoid must be perpendicular to the height. The lateral sides of the trapezoid aren’t perpendicular to either of the bases, so a dotted line is

drawn to represent the height. b1

b2

h

Area of a trapezoid The trapezoid below has a b1 of 14 millimeters, b2 of 20

millimeters, and a height of 22 millimeters. So, the area of this trapezoid is:

A = 1/2 · (b1 + b2) · h

A = 1/2 · (14 mm + 20 mm) · 22 mm

A = 1/2 (34 mm) 22 mm

A = 1/2 · 748 mm2

A = 374 mm2

b1 = 14 mm

b2 = 20 mm

h = 22 mm

This site has a lesson on the area of trapezoids.

Practice Websites These sites have great examples of how to

find the area of polygons and circles. The second one is a worksheet to help you

practice your skills. It is a good idea to take a look at them and see if you learn more

about area. Then you can take the quiz that it provides. After you do that, you can move

on to my worksheet for more practice.

Area of Polygons and CirclesWorksheet on Area of Polygons and Circles

Worksheet Time Now, that you have went through my lesson about areas of polygons.

So, you can apply what you learned with this worksheet. Time

to practice!

Here are the answers to my worksheet.