# Pentagon – Definition With Examples

Updated on January 1, 2024

One of the shapes your kids will encounter as they learn geometry is a shape with 5 sides called the pentagon. They will learn the pentagon definition, how to draw it, calculate the area of the pentagon and the perimeter, and how to use the knowledge on the topic to solve real-life problems. If you want to teach your kids geometry, you have to learn everything about pentagons. Let this article teach you the basics of the pentagon concept so you can pass the information to your kids.

## What Is a Pentagon?

A pentagon is a 5 sided shape the name of which originated from the Greek word Pentegonia. Pente means 5, and gonia means angles. A pentagon’s sides and angles are usually the same as long as it’s a regular pentagon. Each interior angle in a pentagon is 108 degrees, adding up to 540 degrees. On the other hand, the exterior angles of a regular pentagon are 74 degrees. If you want to draw a pentagon, draw a circle and then use a compass to divide it into 5 equal parts.

## Types of Pentagons

There are different types of pentagons, but the two main ones are regular and irregular pentagons.

### Regular pentagon

A regular pentagon is the one where all the sides and angles are equal. It has both rotational symmetry and reflection symmetry. You may be asked, “What does a pentagon look like?” But you may not readily have the answers because there are so few real-life representations of that shape.

An example of a pentagon is the design of the US Department of Defence building, which coincidentally is called the Pentagon. You can also find a regular pentagon in the shape of a home plate on a baseball diamond.

### Irregular pentagon

In an irregular pentagon, all the pentagon sides and angles are not the same. Some will be different because the side and angles of an irregular pentagon are not congruent. The irregular pentagon is hard to find in natural-life objects, but you can find them in kites, in the shape of a starfish with its five arms, and in some stop signs.

Some other pentagons which your kid will be learning about as they progress through the curriculum include:

- Convex pentagon
- Concave pentagon
- Equilateral pentagon
- Equiangular pentagon
- Golden pentagon
- Self-intersecting pentagon

## Classification of Pentagons Based on Their Angles

You can classify pentagons based on their angles’ measure; and following this classification, there are three of them:

### Acute pentagon

In an acute pentagon, all interior angles measure less than 90 degrees. Therefore, the sum of an acute pentagon’s interior angles is less than 450 degrees.

### Obtuse pentagon

In an obtuse pentagon, one angle is greater than 90 degrees, while the rest measures are the same as an acute angle below 90 degrees. However, the sum of the interior angles of this pentagon will still be 540 degrees, regardless of the difference in measures.

### Reflex pentagon

In a reflex pentagon, one of the angles measures higher than 180 degrees, while the rest go between acute and obtuse pentagon angles. Albeit the angle difference, they all sum up to less than 360 degrees, and all the interior angles are still 540 degrees.

## Properties of a Pentagon

Here are some of the pentagon’s properties:

- A pentagon has 5 sides and 5 angles, and they are all equal; if the pentagon is regular, that is the case. If the pentagon is irregular, some sides and the angles may have different measures.
- When measuring the interior angles of a regular pentagon, you’ll get 540 degrees, with each angle measuring 180 degrees. An irregular pentagon will have different angles.
- Each angle of the exterior of a pentagon measures 72 degrees, and the sum of all of them is 360 degrees.
- A pentagon has 5 diagonals. Diagonals are lines dividing the pentagon across 5 vertices.
- A pentagon has 5 lines of symmetry.
- A regular pentagon has a diagonal-to-side length ratio equal to the golden ratio (1 + sqrt(5))/2, which is approximately 1.618.
- If you draw the diagonals of a regular pentagon, they will divide the shape into five identical triangles.
- A regular pentagon can be inscribed inside a circle.
- A golden ratio is between the length of a diagonal and the length of one side of the pentagon. This ratio is 1.618.
- A circumscribed circle is a circle that goes through all the vertices of a pentagon.
- The center of a circumscribed circle in a pentagon is called an incenter.
- Use a formula to find the distance between two parallel sides of a regular pentagon. The formula is w = s(√5 – 1)/2, where “s” stands for the length of one of the sides.
- A pentagram is a star shape that can be drawn inside the polygon.

## Pentagon Sides

The five straight sides of a pentagon connect 5 vertices or corners. The width of a pentagon shows how far apart its two straight sides are. The pentagon has 5 sides which are all the same length if you are working with a regular pentagon. An irregular pentagon may have different measures of some of the sides. If it’s a regular pentagon (which means all the sides are of the same length), you can find the width by multiplying the length of one side by a unique number (about 0.62) in a formula that you can remember as (√5 – 1)/2.

## Pentagon Angles

To find the total of all interior angles in a regular pentagon, multiply 180° by 2 less than the number of sides (5-2 = 3). The total is 540 degrees.

To calculate the measure of each interior angle in a regular pentagon, divide the total of all interior angles by the number of angles, which is 5. Therefore, each interior angle in a regular pentagon measures 108 degrees.

Importantly, all interior angles in a regular pentagon are equal and measure 108 degrees.

## Perimeter of a Pentagon

The perimeter of a pentagon is the total length of all the shape’s sides. If you want to get the perimeter of a regular pentagon, all you have to do is multiply the length of one side by 5. For an irregular pentagon, you must add each side individually to get the eventual sum.

## Area of a Pentagon

The area of a pentagon shows how much space it takes up inside. To find the area of a regular pentagon, you can use a special formula that involves the length of one side of the shape. The formula is A = (5/4) × s^2 × (√(5 + 2√5)). You can still use the same formula when calculating the area of an irregular triangle. The side lengths may differ, but the principles remain the same.

## Solved Examples on Pentagons

Here are some solved examples on pentagons:

### Example 1

Find the perimeter of an irregular pentagon if the sides are 5 cm, 8 cm, 7 cm, 9 cm, and 6 cm.

Solution: We can add up the length of each side to find the perimeter. Therefore, the perimeter is 5 cm + 8 cm + 7 cm + 9 cm + 6 cm = 35 cm.

### Example 2

Find the area of an irregular pentagon if the sides are 6 cm, 7 cm, 6 cm, 8 cm, and 9 cm, and the perpendicular distance from the shortest side to the opposite vertex is 5 cm.

Solution: We can divide the irregular pentagon into a rectangle and a triangle. The rectangle has a base of 6 cm and a height of 5 cm, so its area is 6 cm × 5 cm = 30 cm^2. The triangle has a base of 8 cm and a height of 5 cm, so its area is 8 cm × 5 cm × 1/2 = 20 cm^2. Therefore, the area of the irregular pentagon is 30 cm^2 + 20 cm^2 = 50 cm^2.

### Example 3

Find the length of the apothem of a regular pentagon if each side has a length of 10 cm.

Solution: We can use the formula for the apothem length, which is a = s × (√5 – 1)/4. Substituting the values, we get a = 10 × (√5 – 1)/4 ≈ 3.090 cm.

## Frequently Asked Questions

### What are the angles of a pentagon?

To find the total of all interior angles, multiply all the angles by 3, which is 180*3, and that will give you 540 degrees. Then, divide 540 by 5 to get the size of each angle.

### How many sides does a pentagon have?

A pentagon has 5 sides.

### How many types of pentagons are there?

There are two major types of pentagons, regular and irregular. Then, there are other pentagons that kids learn eventually.

### What is the sum of all interior angles of a pentagon?

The sum of all the interior angles of a pentagon is 540 degrees.

### How many diagonals does a pentagon have?

A pentagon has 5 diagonals.

### Does a pentagon always have equal sides?

A regular pentagon has equal sides and angles. However, the sides’ length and the angles’ measure may change in an irregular pentagon.