# Pentagon

## Definition

A pentagon is a closed and flat geometrical shape having five straight sides and five angles. The word ‘pentagon’ came from the Greek word ‘pente’ meaning ‘five’ and ‘gonia’ meaning ‘angle’.

When all the five sides and angles of a pentagon are equal, it is called a regular pentagon. Otherwise, it is an irregular pentagon. In our article, we deal with only regular pentagons to better understand the concepts regarding the shape.

## Properties

1. Has five sides of equal length; in ⌂ ABCDE, AB = BC = CD = DE = EA
2. Has five interior angles, each measuring 108°; so ∠ABC =∠BCD = ∠CDE = ∠DEA = ∠EAB = 108°
3. The sum of the interior angles is 540°; so ∠ABC +∠BCD + ∠CDE + ∠DEA + ∠EAB = 540°
4. Has five exterior angles, each measuring 72°
5. Has five diagonals; shown as AC, AD, BD, BE, and CE

## Formulas

### Perimeter

It is the total distance covered around the edge of the pentagon. The formula is given below:

Perimeter (P) = 5s, here s = side length

Find the perimeter of a regular pentagon whose each side measure 15 cm.

Solution:

As we know,
Perimeter (P) = 5s, here s = 15 cm
= 5 x 15 cm
= 75 cm

### Area

The total space enclosed by the pentagon. The formula is given below:

Area (A) = ½(p x a), here p = perimeter, a = apothem

Find the area of a pentagon whose perimeter is 40 cm and apothem length is 9 cm.

Solution:

As we know,
Area (A) = ½(p x a), here p = 40 cm and a = 9 cm
= ½ (40 x 9) cm2
= 180 cm2

Problem: Finding the area of a regular pentagon when the SIDE and APOTHEM are known

Find the area of a pentagon whose side length is 6 cm and apothem length is 8 cm.

Solution:

As we know,
Area (A) = 5/2(s x a), here s = 6 cm and a = 8 cm
= 5/2 (6 x 8) cm2
= 120 cm2

## Types

Depending on the sides, angles, and vertices, pentagon shapes are classified into the following types:

1. Regular Pentagon: Have five sides of equal length and five interior angles each measuring 108° and exterior angles of 72°each. A regular pentagon are thus both equilateral (having five equal sides) and equiangular (having five angles of equal measure). Drawing diagonal lines between the nonadjacent vertices of a regular pentagon results in a star shape known as pentagram. A regular pentagon has five lines of symmetry.
2. Irregular Pentagon: Does not have all sides equal or all angles equal and thus have no specific angles.
3. Convex Pentagon: Have all vertices pointing outwards and thus the angles formed are less than 180°. A convex pentagon thus can be regular.
4. Concave Pentagon: Have at least one vertex pointing inwards whose angle is greater than 180°. Thus all concave pentagons are irregular.

## Real Life Examples

• The famous U.S. department of defense building in Washington D.C popularly known as the Pentagon building
• The home plate in baseball (changed to pentagon from square to minimize accidents due to sharp corners)
• Gynoecium of an apple containing five carpels
• Star fruit with a five-fold symmetry
• The shape of Echinoderms such as a sea star
• The endoskeleton of a sea urchin
• More Resources