Understanding Angles in a Pentagon

A pentagon is a fascinating two-dimensional shape with five sides and five angles. It is important to note that for a shape to be a pentagon, its five sides must be connected and have no curved sides. This unique shape can be found in various real-life examples, such as the black sections on soccer balls, school crossing signs, and even the famous Pentagon building in the US.

Moreover, pentagons can also be spotted in nature, like in flowers or the cross-sections of okra. In this article, we will delve deeper into the different types of pentagons and explore the angles within this intriguing shape.

Different Types of Pentagons

Pentagons can be classified into various types based on their properties. Here are the main categories:

• Regular Pentagon: When all sides and interior angles of a pentagon are equal, it becomes a regular pentagon.
• Irregular Pentagon: If the sides of a pentagon are not equal and the angles are not of the same measure, it is classified as an irregular pentagon.
• Convex Pentagon: This type of pentagon has vertices pointing outwards, and all its interior angles are less than 180°.
• Concave Pentagon: If any one of the interior angles in a pentagon is more than 180° and the vertices point inward, the shape is known as a concave pentagon.

Take a look at the image below to see the visual representation of these different types of pentagons.

Types of Pentagons

Understanding Pentagon Definition

Before we delve into the angles within a pentagon, let's start with a clear definition. A pentagon is a geometric two-dimensional shape with five sides and five angles. The word "pentagon" is derived from the Greek words "penta," which means five, and "gon," which translates to angle.

To provide a real-life example, the shape of a home plate seen on a baseball field is a great representation of a pentagon. Now that we understand the definition let's explore the angles within a pentagon.

Interior Angles

In a regular polygon, an interior angle is an angle inside the shape, formed between two joined sides. The total number of interior angles in any polygon is equal to the total number of sides. In the case of a pentagon, there are five interior angles.

Each interior angle of a regular pentagon can be calculated using the following formula:

Each interior angle = [(n - 2) × 180°] / n

Where 'n' represents the number of sides. In the case of a pentagon, n equals 5. By substituting this value into the formula, we can calculate that each interior angle of a regular pentagon equals 108°.

Interior Angles of a Pentagon

Exterior Angles

When a side of a pentagon is extended, the angle formed outside the pentagon with its side is called an exterior angle. In a regular pentagon, each exterior angle is equal to 72°. The sum of the exterior angles of any regular pentagon is always 360°.

The formula for calculating the exterior angle of a regular polygon is:

Exterior angle of a regular polygon = 360° / n

Keep in mind that 'n' represents the total number of sides in a pentagon. Take a look at the image below to visualize the exterior angles of a pentagon.

Exterior Angles of a Pentagon

Central Angles

The center of a pentagon is the point equidistant from each vertex or corner. When we join this center point with all the vertices, we create five central angles at the center. To calculate the measure of the central angle of a regular pentagon, we have two methods:

Method 1:

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

Method 2:

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

Central Angles in a Pentagon

Sum of Interior and Exterior Angles

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's explore how to calculate the sum of the 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, divide it into triangles. As depicted in the image below, three triangles can be formed in a pentagon. The sum of the angles in each of these triangles is 180°. Therefore, to calculate the sum of the interior angles of the pentagon, multiply the sum of the angles in each of these triangles by the total number of triangles. This results in: 180° × 3 = 540°. Hence, the sum of the interior angles of a pentagon is equal to 540°.

Sum of the Interior Angles of a Pentagon

Another way to calculate the sum of the interior angles of a pentagon is by using the formula:

Sum of angles = (n - 2) × 180°

Here, 'n' represents the number of sides of the polygon. Substituting the value of 'n' (which is 5) into the formula, we get: (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 always equal to 360°. Let's prove this through the following steps:

1. The sum of interior angles of a regular polygon with 'n' sides is given by the formula: 180°(n-2).
2. Each exterior angle is supplementary to the interior angle. Therefore, each exterior angle will be: [180n - 180(n-2) + 360] / n = 360 / n.
3. Thus, the sum of the exterior angles will be: n(360 / n) = 360°. Hence, the sum of exterior angles of a pentagon is 360°.

Important Notes

To summarize the key points regarding 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 is 540°, while the sum of all the exterior angles is 360°.
• Each exterior angle of a regular pentagon measures 72°, and each interior angle measures 108°.

By understanding the angles within a pentagon, we gain a deeper appreciation for the beauty and intricacy of this shape.