Math resources Geometry Area

Area of a pentagon

# Area of a pentagon

Here you will learn about the area of a pentagon, including how to find the area of a pentagon made from a triangle and a rectangle and how to find the area of a regular pentagon.

Students will first learn about the area of a pentagon as a part of geometry in high school.

## What is the area of a pentagon?

The area of a pentagon is the amount of space inside a 2D, five-sided polygon. A regular pentagon is a type of pentagon that has equal sides and all angles have an equal measure.

The area of a regular polygon can be found by splitting the shape into congruent isosceles triangles. You can find the area of one of the triangles and then multiply by the number of sides to find the total area of the regular polygon.

To find the area of a regular pentagon, you can use the following formulas,

The area of a regular pentagon can be calculated by:

\text {Area of a regular pentagon }=5 \times \text { area of one triangle. }

The formula for the area of a regular pentagon can also be written as

\text {Area of a regular pentagon }=5 \times \cfrac{1}{2} \, b h.

Where b is the base length (or the side of the pentagon) and h is the perpendicular height of the triangle. Sometimes these values will need to be calculated.

For example,

The area of this regular pentagon is

\begin{aligned}& =5 \times \cfrac{1}{2} \times 12 \times 5 \\\\ & =150 \mathrm{~cm}^2 \end{aligned}

### Area of an irregular pentagon

An irregular pentagon may have been formed by combining other shapes such as rectangles and triangles. The area of each of these can be found and added together to find the area of the pentagon.

For example,

The area of irregular pentagon ABCDE is

\text{Total area }=(\cfrac{1}{2}\times 10\times 5)+(4\times 10)=25+40=65 \ cm^2.

### Angles in a regular pentagon

You can find the angles of a regular polygon to answer a variety of different area questions.

Splitting the regular pentagon into 5 congruent isosceles triangles, use the angle fact that the sum of internal angles at a point is 360^{\circ} to find the angle at the top of the isosceles triangle.

360\div 5=72^{\circ}

The equal base angles can be found by using the angle fact that the sum of angles in a triangle is 180^{\circ}.

(180-72)\div 2=54^{\circ}

Step-by-step guide: Types of angles

### The radii of a regular pentagon

A regular pentagon has five radii. The apothem is the radius of the inscribed circle, R_i.

The radius of the inscribed circle is perpendicular to the sides of the regular pentagon and meets the side at the midpoint.

When a regular pentagon is drawn within a circle the circle is known as the circumcircle and the radius is R_c.

Area formula using the perimeter and apothem of a pentagon

The height (or inscribed radius) in this regular pentagon is also known as the apothem of the pentagon.

We can use the following formula to find the area of a regular pentagon,

\text{Area of a regular pentagon }= \cfrac{1}{2}\times \text{ perimeter }\times \text{ apothem}.

The perimeter of the pentagon can be found by multiplying the side length by 5.

## Common Core State Standards

How does this relate to high school math?

• High School: Geometry (HS.G.SRT.D.11)
Understand and apply the Law of Sines and the Law of Cosines to find unknown measurements in right and non-right triangles (e.g., surveying problems, resultant forces).

## How to find the area of a pentagon

In order to find the area of a pentagon, you will:

1. Consider the pentagon as a compound shape and split it appropriately.
2. Calculate the area of the parts.
3. Calculate the total area of the pentagon.
4. Write the answer, including the correct units.

## Area of a pentagon examples

### Example 1: area of an irregular pentagon formed with one triangle and a rectangle

ABCDE is a pentagon.

Find the area of the pentagon.

1. Consider the pentagon as a compound shape and split it appropriately.

The diagram given in the question is already split into a rectangle and a triangle.

2Calculate the area of the parts.

Find the area of the triangle ABC.

The height of the triangle, h, is h=8-5=3.

Therefore, the area of the triangle is A=\cfrac{1}{2} \, bh=\cfrac{1}{2}\times 10\times 3=15.

The area of the rectangle ACDE is A=bh=10\times 5=50.

3Calculate the total area of the pentagon.

The total area of the pentagon is found by adding the area of the triangle and the area of the rectangle together.

\text{Total Area }=15+50=65

4Write the answer, including the correct units.

The dimensions of the shape are given in meters, so the units of the area will be in square meters. The area of pentagon ABCDE is 65 \, m^2.

### Example 2: area of an irregular pentagon formed with one isosceles triangle and a rectangle

ABCDE is a pentagon.

Calculate the area of the pentagon.

Consider the pentagon as a compound shape and split it appropriately.

Calculate the area of the parts.

Calculate the total area of the pentagon.

Write the answer, including the correct units.

### Example 3: calculating the area of a regular pentagon given the area of one of the isosceles triangles

Calculate the area of the regular pentagon ABCDE given that the area of one of the congruent triangles is 20 \, cm^2.

Consider the pentagon as a compound shape and split it appropriately.

Calculate the area of the parts.

Calculate the total area of the pentagon.

Write the answer, including the correct units.

### Example 4: area of a pentagon given the side length

Calculate the area of the regular pentagon ABCDE given that the length of one side of the regular pentagon is 10 \, cm.

Consider the pentagon as a compound shape and split it appropriately.

Calculate the area of the parts.

Calculate the total area of the pentagon.

Write the answer, including the correct units.

### Example 5: area of a pentagon given the height (apothem)

Calculate the area of the regular pentagon ABCDE given that the length of the apothem of the regular pentagon is 12 \, cm.

Consider the pentagon as a compound shape and split it appropriately.

Calculate the area of the parts.

Calculate the total area of the pentagon.

Write the answer, including the correct units.

### Example 6: area of a pentagon given the radius of the circumcircle

Calculate the area of the regular pentagon ABCDE given that the length from the center of the pentagon to the vertices is 15 \, cm.

Consider the pentagon as a compound shape and split it appropriately.

Calculate the area of the parts.

Calculate the total area of the pentagon.

Write the answer, including the correct units.

### Example 7: area of a pentagon given the radius of the circumcircle (diagonals)

Consider the pentagon as a compound shape and split it appropriately.

Calculate the area of the parts.

Calculate the total area of the pentagon.

Write the answer, including the correct units.

### Teaching tips for the area of a pentagon

• Start by breaking down the formula for the area of a regular pentagon. This isn’t as commonly memorized as some other formulas, so students will benefit from the additional instruction.

• Instead of giving students worksheets to work on independently, consider allowing students to work through these practice problems in small groups or in pairs. This allows students to discuss and collaborate on answers, which can lead to students learning from one another.

• Use technology, such as an online pentagon calculator or geometry software, which allows learning to be more interactive and engaging for some students.

### Easy mistakes to make

• Using the wrong units
Remember that area is measured in square units. Square units include square centimeters (cm^2), square meters (m^2) and square millimeters (mm^2).

• Confusing regular and irregular pentagons
It’s important to identify whether a pentagon is regular or irregular before calculating the area. Remember regular pentagons have all equal sides, while irregular pentagons do not.

### Practice area of a pentagon questions

1. Calculate the area of the irregular pentagon ABCDE given that ACDE is a rectangle.

1050 \, m^2

360 \, m^2

255 \, m^2

202.5 \, m^2

The area of the triangle ABC is A=\cfrac{1}{2} \, bh=\cfrac{1}{2}\times 15\times 7=52.5.

The area of the rectangle ACDE is A=bh=15\times 10=150.

The total area of the pentagon is found by adding the area of the triangle and the area of the rectangle together.

\text{Total Area}=52.5+150=202.5 \ m^2

2. Calculate the area of the irregular pentagon ABCDE given that ABC is an isosceles triangle and ACDE is a rectangle.

1580 \, m^2

1280 \, m^2

1780 \, m^2

1880 \, m^2

The height of the triangle is h=\sqrt{25^2-20^2}=15.

The area of the triangle ABC is A=\cfrac{1}{2} \, bh=\cfrac{1}{2}\times 40\times 15=300.

The area of the rectangle ACDE is A=bh=40\times 32=1280.

The total area of the pentagon is found by adding the area of the triangle and the area of the rectangle together.

\text{Total Area}=300+1280=1580 \ m^2

3. Calculate the area of the regular pentagon ABCDE given that the area of one of the congruent triangles is 15 \, cm^2.

3 \, cm^2

60 \, cm^2

75 \, cm^2

100 \, cm^2

The area of the regular pentagon is

\begin{aligned} & =5 \times \text { area of one triangle } \\\\ & =5 \times 15 \\\\ & =75 \mathrm{~cm}^2 \end{aligned}

4. Calculate the area of the regular pentagon ABCDE given that the side length is 20 \, cm.

1453 \, cm^2

951 \, cm^2

688 \, cm^2

138 \, cm^2

The height of the triangle is

\text{height}=\tan(54)\times 10=13.7638…

The area of one of the isosceles triangles is

\begin{aligned}A&=\cfrac{1}{2} \, bh \\\\ &=\cfrac{1}{2}\times 20\times 13.7638… \\\\ &=137.638… \end{aligned}

The area of the regular pentagon is

\begin{aligned} & =5 \times \text { area of one triangle } \\\\ & =5 \times 137.638 \\\\ & =688.190 \ldots \end{aligned}

The area of the regular pentagon is 688 \, cm^2 to 3 significant figures.

5. Calculate the area of the regular pentagon ABCDE given that the distance from the center of the pentagon to the midpoint of one of the sides is 40 \, cm.

2750 \, cm^2

5810 \, cm^2

11011 \, cm^2

3804 \, cm^2

Using trigonometry,

x=40\div \tan(54)=29.0617…

The base of one of the isosceles triangles is

\text{base}=2x=2\times 29.0617…=58.1234…

The area of one of the isosceles triangles is

\begin{aligned}A&=\cfrac{1}{2}bh. \\\\ &=\cfrac{1}{2}\times 58.1234…\times 40 \\\\ &=1162.468… \end{aligned}

The area of the regular pentagon is

\begin{aligned} & =5 \times \text { area of one triangle } \\\\ & =5 \times 1162.468 \ldots \\\\\ & =5812.34 \ldots \end{aligned}

The area of pentagon ABCDE is 5810 \, cm^2 (to 3 significant figures).

6. Calculate the area of the regular pentagon ABCDE given that the distance between the center of the pentagon and one of the vertices is 17 \, cm.

497 \, cm^2

1050 \, cm^2

687 \, cm^2

137 \, cm^2

Using trigonometry,

The height of the triangle can be calculated using trigonometry and the angles in the isosceles triangle. The hypotenuse is 17 \, cm.

\text{height}=\sin(54)\times 17=13.7532…

You need to work out the base of the triangle using trigonometry.

x=\cos(54) \times 17=9.9923…

The base of one of the isosceles triangles is

\text{base}=2x=2\times 9.9923…=19.9846…

The area of one of the isosceles triangles is

\begin{aligned}A&=\cfrac{1}{2}bh. \\\\ &=\cfrac{1}{2}\times 19.9846…\times 13.7532… \\\\ &=137.427… \end{aligned}

The area of the regular pentagon is

\begin{aligned} & =5 \times \text { area of one triangle } \\\\ & =5 \times 137.427 . . \\\\ & =687.138 \ldots \end{aligned}

The area of pentagon ABCDE is 687 \, cm^2 (to 3 significant figures).

## Area of a pentagon FAQs

What is the area of a pentagon formula?

A formula that can be used to find the area of a pentagon is

\text{Area of a regular pentagon } = 5 \, \times \text{ area of one triangle,} or

\text{Area of a regular pentagon } = 5 \times \cfrac{1}{2} \, bh.

How do you find the measure of the apothem?

The apothem is the distance from the center of a polygon to its sides. To find the apothem, find the center of the shape and then draw a line from the center to the midpoint of one side of a pentagon.

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