Angles in a Pentagon

A pentagon is a two-dimensional polygon with five sides and five angles. If the five sides of a shape are not connected, or if the shape has a curved side, then it is not a pentagon. Some of the real-life examples of a pentagon are the black sections on soccer balls, school crossing signs, the Pentagon building in the US, and so on. This shape can also be spotted in flowers and even in the cross-sections of okra.

Let us learn more about the angles in a pentagon in this page.

Pentagons can be categorized into different types based on their properties. Here is a list of the types of pentagons classified according to the sides, angles and vertices:

• A pentagon becomes a regular pentagon when all its sides and interior angles are equal.
• If the sides of a pentagon are not equal and the angles are not of the same measure, it is an irregular pentagon.
• A convex pentagon has vertices pointing outwards and its interior angles are less than 180°.
• If any one of the interior angles in a pentagon is more than 180° and if the vertices point inwards, then the pentagon is concave.

Pentagon Definition

A pentagon is defined as a geometric two-dimensional shape with five sides and five angles. The word of the shape is derived from the Greek word as “Penta” denotes five, and “gon” denotes angle. The pentagon is a 5-sided polygon and a real-life example of a pentagon shape is the home plate seen on a baseball field.

A pentagon is a two-dimensional polygon with five angles. An angle is formed when two sides of the pentagon share a common endpoint, called the vertex of the angle. In this section, let us learn about the kinds of angles, like, the interior angles, exterior angles, and central angles.

Interior Angles:

In a regular polygon, an angle inside the shape, between two joined sides is called an interior angle. For any polygon, the total number of interior angles is equal to the total number of sides. In a pentagon, there are five interior angles. Each interior angle of a regular pentagon can be calculated by the formula: Each interior angle = [(n – 2) × 180°]/n ; where n = the number of sides. In this case, n = 5. So, substituting the value in the formula: [(5 – 2) × 180°]/5 = (3 × 180°)/5 =108°

Observe the following pentagon which shows that each interior angle of a regular pentagon equals 108°.

Exterior Angles:

When the side of a pentagon is extended, the angle formed outside the pentagon with its side is called the exterior angle. Each exterior angle of a regular pentagon is equal to 72°. The sum of the exterior angles of any regular pentagon equals 360°. The formula for calculating the exterior angle of a regular polygon is: Exterior angle of a regular polygon = 360° ÷ n. Here, n represents the total number of sides in a pentagon. Observe the following figure which shows the exterior angles of a pentagon.

Central Angles:

The center of a pentagon is the point that is equidistant from each vertex or corner. The central angles of any pentagon are formed when this center point is joined to all the vertices, resulting in 5 central angles at the center. There are two ways to find the measure of the central angle of a regular pentagon.

Method 1:The following steps can be followed to find the measurement of the central angles:

• Step 1: In the following pentagon ABCDE, mark the center as O and join the center O to the vertices A,B,C,D, and E, forming 5 triangles.
• Step 2: Since the center is equidistant from all the vertices, and all the sides of a regular pentagon are equal, all these triangles will be isosceles and congruent to each other. We can thus conclude that all 5 angles at the center will be equal.
• Step 3: We know, that all the interior angles of a pentagon measure 108°. Since the triangles are congruent, the interior angle at each vertex will be bisected to equal halves, each measuring (108°/2) = 54°.
• Step 4: Apply the angle sum property of a triangle to find the central angle. Using this we can calculate the measurement of each central angle as: Central angle of a regular pentagon = 180° - (2 × 54°) = 72°

Method 2: The following steps can be followed to calculate the central angle of a pentagon under this method:

• Step 1: Mark the centre of the pentagon and draw congruent triangles as shown in the previous method to get 5 equal angles resulting from the division of the central angle.
• Step 2: Since all the five angles in the centre are equal, we can get the value of each angle: 360° ÷ 5 = 72°.
• Step 3: Hence, the central angle in a regular pentagon measures 72°.

The sum of the angles in any polygon depends on the number of sides it has. In the case of a pentagon, the number of sides is equal to 5. Let us see how to calculate the sum of interior and exterior angles in a pentagon.

Sum of Interior Angles in a Pentagon

To find the sum of the interior angles of a pentagon, we divide the pentagon into triangles. Observe the following figure which shows that three triangles can be formed in a pentagon. The sum of the angles in each of these triangles is 180°. So, in order to get the interior angles of this pentagon, we multiply the sum of the angles of each of these triangles with the total number of triangles. This makes it: 180° × 3 = 540°. Hence, the sum of the interior angles of a pentagon is equal to 540°.

Another way to calculate the sum of the interior angles of a pentagon is by using the formula: Sum of angles = (n – 2)180°; where 'n' represents the number of sides of the polygon. Substituting the value of 'n' in the formula: (5– 2)180° = 540°. Therefore, the sum of the interior angles of a pentagon is 540°.

Sum of Exterior Angles in a Pentagon

The sum of exterior angles of a polygon is equal to 360°. Let us prove this now with the following steps:

• The sum of interior angles of a regular polygon with 'n' sides = 180 (n-2).
• Hence, each interior angle is: 180 (n-2)/n.
• We know that each exterior angle is supplementary to the interior angle, so, each exterior angle will be: [180n -180n + 360]/n = 360/n.
• Now, the sum of the exterior angles will be: n (360/n)= 360°. Hence, the sum of exterior angles of a pentagon equals 360°.

Important Notes

Here is a list of a few points that should be remembered while studying about the angles in a pentagon:

• A pentagon is a two-dimensional polygon with five angles and five sides.
• The sum of all the interior angles of any regular pentagon equals 540° and the sum of all the exterior angles of any regular pentagon equals 360°.
• Each exterior angle of a regular pentagon is equal to 72° and each interior angle of a regular pentagon is equal to 108°.

Related Topics

• Angles
• Triangles
• Pentagon Shape
• Definition of Polygon
• Closed Shapes
• Pentagonal Prism
• Perimeter of a Triangle

What is the Sum of All the Interior Angles in a Pentagon?

The sum of all the interior angles of a regular polygon can be calculated by the formula: Sum of angles = (n – 2)180°, where 'n' represents the number of sides. Substituting the value of 'n' in the formula: (5 – 2)180° = 540°. Therefore, all the 5 angles in the pentagon sum up to 540°.

Does a Pentagon Have a 90° Angle?

Pentagons can have a maximum of three 90° angles. Therefore, pentagons can have a 90° angle.

Are All Angles in a Pentagon Equal?

If the pentagon is a regular pentagon, then all its angles are equal. However, if the pentagon is not a regular one, then the measure of all the angles will be different.

Why Does a Pentagon Have No Parallel Lines?

A regular pentagon does not have any parallel lines. However, if the pentagon is an irregular pentagon, then one pair (2 parallel lines) or two pairs (4 parallel lines) of lines can be parallel.

How Do You Calculate the Angles in a Pentagon?

The measure of angles in any polygon can be calculated using different formulas depending upon the type of angle. For example, the interior angle of a polygon can be calculated using the formula: Measure of each angle = [(n – 2) × 180°]/n, where 'n' is number of sides (5 for a pentagon). Therefore, after substituting the value of 'n' in this formula, we find the measure of an interior angle in a pentagon to be 108°. The formula to calculate each exterior angle of a polygon is: Exterior angle = 360°/n. For a pentagon, n = 5. Hence, each exterior angle of a pentagon measures 360°/5 = 72°.

What is a Pentagonal Prism?

A pentagonal prism can be defined as a three-dimensional solid that has two pentagonal bases at the top and bottom and five rectangular sides.