## Pentagon Definition

In geometry, a pentagon is a five-sided polygon with five straight sides and five interior angles that sum up to $540°$. A pentagon shape is a plane figure, or flat (two-dimensional) 5-sided geometric shape.

### Properties of a Pentagon

Pentagons can be simple or self-intersection. The properties of a simple pentagon (5-gon) are it must have five straight sides that meet to create five vertices, but do not self-intersect:

1. Pentagons have five straight sides
2. Pentagons have five interior angles, which sum to $540°$
3. The five sides do not intersect
A self-intersecting regular pentagon is called a pentagram.

## Types of Pentagons

The two types of pentagons in geometry are regular pentagons and irregdular pentagon.

### Regular Pentagons

A regular pentagon must have five congruent sides, five congruent interior angles, and congruent exterior angles:

• five congruent sides (sides of equal length)
• five congruent interior angles (each measuring $108°$)
• five congruent exterior angles of $72°$
Regular pentagons do not have parallel sides.

Just like any regular polygon, going around a pentagon completes one full circle, so the exterior angles are found by dividing $360°$ by the number of sides, in this case, .

### Irregular Pentagons

Irregular pentagons can be a convex pentagon or a concave pentagon, but they must have five different-length sides.

• Convex pentagon - No internal angles can be more than $180°$
• Concave pentagon - One interior angle that is greater $180°$

A common example of a convex irregular pentagon is the home plate on a baseball field.

All pentagons (regular and irregular) are five-sided shapes, with five interior angles and five exterior angles.

## Examples of Pentagons

If you look for a pentagon shape around you, you will surely find it. Whether it is an irregular pentagon with varying side lengths or a regular pentagon with equal sides and equal angles, there are many real-life examples of pentagons:

• The famous U.S. department of defense building in Washington D.C. (The Pentagon building)
• The home plate on a baseball field
• School crossing signs
• Sections on a soccer ball

When you find a five-sided shape, you can describe it as a pentagonal shape.

## Perimeter and Area of a Pentagon

The perimeter of a pentagon is the distance around its five straight sides. There is a simple formula to find the perimeter of a regular pentagon if you know one side length.

To find the perimeter of an irregular pentagon, you must measure and add up the five sides.

A line segment drawn from the center of a regular pentagon perpendicular to a side is called the apothem. The apothem is used to calculate the area of a pentagon.

The area, $A$, of a pentagon is the space inside it's five straight sides. How to find the area will depend on which type of pentagon you have and what information you know about your pentagon.

## Lesson Summary

Now that you've completed this lesson. You should know:

• That a pentagon is a five-sided polygon with five angles
• The identifying properties of a pentagon
• The different types of pentagons
Instructor: Malcolm M.
Malcolm has a Master's Degree in education and holds four teaching certificates. He has been a public school teacher for 27 years, including 15 years as a mathematics teacher.

## Regular Pentagon

In geometry, a pentagon is a five-sided polygon or 5-gon.

• Number of sides: 5
• Number of vertices: 5
• Interior angle: $108°$; summing to $540°$
• Exterior angle: $72°$; summing to $360°$
• Area: ½ × perimeter x apothem (a)
• Perimeter: 5 × side
• Properties: Convex, cyclic, equilateral, isogonal, isotoxal
• Type: Regular polygon

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