Angles in a Pentagon

Pentagon is a two-dimensional (2D) five-sided polygon having five angles. In a regular pentagon, the sum of all interior angles is equal to 540° with each interior angle measuring 108°, and each exterior angle measuring 72°.

In this article, we will discuss about what is a pentagon, different types of Pentagon, various angles in the Pentagon along with some solved questions.

What is a Pentagon?

Pentagon is classified as a two-dimensional, flat, or plane figure with five sides. These sides are interconnected, forming a closed shape. The word “Penta” means five and “gon” means angles. Hence, a Pentagon is characterized by having exactly 5 sides and five angles. The sum of all interior angles in a regular pentagon is equal to 540°.

Types of Pentagon

A Pentagon can be classified into various types:

• Regular Pentagon: A regular pentagon is a pentagon whose all the five sides and all the five angles are equal.

• Irregular Pentagon: In case of an irregular pentagon, neither the sides nor the angles are equal.

• Convex Pentagon: In convex pentagons, all the interior angles are less than 180°.

• Concave Pentagon: Any one of the interior angle must be greater than 180°.

Angles in Pentagon

Pentagon is a closed two-dimensional polygon which has 5 angles. When two sides of the pentagon share a common vertex, an angle is formed. There are 3 types of angles in a Pentagon namely:

• Interior Angles
• Exterior Angles
• Central Angles

Let us look and understand all the angles in detail.

Angles in Pentagon

Interior Angles in Pentagon

The angle present inside the pentagon between the two vertex is known as Interior angle of a Pentagon. The total number of Interior angles present inside the pentagon is equal to the total number of sides i.e. 5.

Interior Angle in Regular Pentagon

The measure of an interior angle of a regular polygon with side n can be calculated by:

[(n – 2) × 180°]/n , where n = number of sides

For a regular pentagon which has 5 sides, we can find the value of each internal angle i.e.,

[(5 – 2) × 180°]/5

= (3 × 180°)/5

= 108°

Therefore, each internal angle of a regular pentagon is of 108° each.

Interior Angles of Pentagon

Note: The measure of each interior angle of an irregular pentagon is different but the sum of all interior angle remains same i.e. 540°.

Exterior Angles in Pentagon

When two of the vertices of a pentagon is joined, the angle which is formed outside the pentagon is known as exterior angle. The total number of Exterior angles in the pentagon is equal to the total number of sides i.e. 5. The measure of each exterior angle of a regular pentagon is always same.

Exterior Angle in Regular Pentagon

The measure of each exterior angle of a regular polygon with ‘n’ sides can by calculated by:

360° ÷ n , where n = number of sides

For a regular pentagon which has 5 sides, we can find the value of each external angle i.e.,

360/5

= 72°

Therefore, each external angle of a regular pentagon measures 72° each.

Exterior Angle of a Pentagon

Note: The measure of each exterior angle of an irregular pentagon is different but the sum of all exterior angle always remain same i.e. 360°.

Central Angles in Pentagon

The center of a pentagon is the point that is equidistant from each vertex. The measure of the central angle of a regular pentagon makes a circle, i.e. total measure is 360°.

The steps to find central angle of a pentagon are:

• Step 1: Mark the center of the pentagon and draw congruent triangles by joining each vertex to the centre point to get 5 equal triangles.

• Step 2: Since all the five triangles formed are isosceles and congruent to each other therefore all the five angles formed at the centre by these triangles must be equal.

• Step 3: The angle formed at the center is 360° which is divided into 5 equal angles. Hence, the central angle in a regular pentagon measures 360°/5 = 72°.

Therefore, the central angle inside a regular pentagon is of 72° each.

Central Angle of Pentagon

Sum of Angles in a Pentagon

The sum of the angles in a Pentagon can be classified into 2 different types which are:

• Sum of Exterior Angles
• Sum of Interior Angles

Let us understand each of them.

Sum of Interior Angles

Sum of Interior angle of a pentagon can be found by dividing the pentagon into three triangles. The sum of the angles in each of the triangles inside a pentagon is 180° and there are 3 such triangles. Therefore,

Sum of Interior Angles in a Pentagon = 180° × 3 = 540°.

Sum of Interior Angles of a Pentagon

Also, the sum of the interior angles of a polygon of n sides is given by:

(n – 2) × 180°

(5-2) × 180° (here n = 5)

180° × 3 = 540°.

As we know that the sum of interior angles of a regular polygon with ‘n’ sides = 180 (n-2). Hence, each interior angle is: 180 (n-2)/n.

Sum of Exterior Angles

Sum of exterior angle of every polygon is always 360°. In case of regular polygon with side ‘n’, each exterior angle measures 360/n each. If the polygon is irregular then the measure of each exterior angle will be different but it will sum up to 360° always.

Sum of exterior angles of a Pentagon is equal to 360°.

Each interior angle of a pentagon is measured by 180 (5-2)/5

We know that each exterior angle is supplementary to the interior angle, so, their sum will be 180°,

Therefore, exterior angle = 180° – [180 (5-2)/5]°

= 180° – 108° = 72°

Now, the sum of the exterior angles will be: 5 × 72° = 360°

Hence, the sum of exterior angles of a pentagon equals 360°.

Important Notes

The main important notes of the angles in Pentagon are:

• A pentagon is a closed two-dimensional polygon which is closed and has 5 sides and 5 angles.

• Sum of all interior angles is equal to 540°.

• Each interior angle of a regular pentagon measures 108°.

• Each exterior angle of a regular pentagon measures 72°.

• The central angle inside a regular pentagon is of 72°.

Read More,

• Area of Pentagon
• Pentagonal Pyramid
• Types of Polygon

Angles in Pentagon Solved Examples

Example 1: The angles at 4 vertices of a pentagon measure 90°, 120°, 50°, and 70°. What is the measure of the remaining angle?

Solution:

We know that for a pentagon, the sum of all the interior angles equals 540°

Therefore, 90° + 120° + 50° + 70° + x = 540°

x = 540°−330°

x = 210°

Example 2: Three angles of a pentagon are 80°, 70° and 100°, then the other two angles can be 145° and 145° or 120° and 180°?

Solution:

Sum of three angles = 80° + 70° + 100° = 250°

We know that the sum of all the five angles of a pentagon is 540°.

Sum of the other two angles = 540° – 250° = 290°

Now,

145° + 145° = 290°

120° + 180° = 300°

Hence, the other two angles of a pentagon are 145° and 145°.

Example 3: If an interior angle of a regular pentagon is 9x. Then, what is the value of x?

Solution:

Each interior angle of a regular pentagon is 108°

Hence, 9 x = 108°

x = 12

Practice Questions on Angles in Pentagon

Q1. Honey found the perimeter of his pentagonal ground to be 150 units. How will he find the area of the ground?

Q2. Rohan knows that the sum of interior angles of a polygon is 540°. Can you help him know how many sides does it have?

Q3. The angles at 4 vertices of a pentagon measure 50°, 80°, 100°, and 40°. What is the measure of the remaining angle?

Q4. If the measure of an interior angle of a regular pentagon is 4x. Find the value of x.

Q.: If five angles of a pentagon are 5x + 3, 2x + 4, 3x, 4x + 3, 5x – 2. Find the value of x

Angles in Pentagon: Frequently Asked Questions

What is a Pentagon?

A pentagon is a closed two-dimensional polygon which is closed and has 5 sides and 5 angles.

What is sum of interior angles of a pentagon?

The sum of interior angles of a pentagon is 540°.

What is sum of exterior angles of a pentagon?

The sum of exterior angles of a pentagon is 360°.

What is measurement of central angle of a regular pentagon?

The measurement of the central angle of a regular pentagon is 72°.

What is measure of each interior angle of a regular pentagon?

The measure of each interior angle of a regular pentagon is 108°

What is measure of each exterior angle of a regular pentagon?

The measurement of each exterior angle of a regular pentagon is 72°.

What is formula for the sum of angles of a pentagon?

The sum of the interior angles of a polygon of n sides = (n – 2) × 180°. Here n = 5, therefore, (5-2) × 180° = 540°