A pentagon is a five-sided polygon that can be found in various forms, such as regular or irregular. In this article, we will explore angles in a pentagon, focusing on the regular pentagon where all sides are equal in length and all angles are equal in measure.
We will show you how to calculate the sum of interior angles, the measure of each angle, and provide some examples for better understanding.
Calculating the Sum of Interior Angles in a Pentagon
Step 1: Recall the polygon angle-sum formula
The polygon angle-sum formula allows us to find the sum of the interior angles of any polygon, regardless of its shape or size. The formula is:
Sum of interior angles = (n – 2) × 180°
where “n” is the number of sides in the polygon.
Step 2: Apply the formula to a pentagon
Since a pentagon has five sides, we can substitute n = 5 into the formula:
Sum of interior angles = (5 – 2) × 180°
Step 3: Solve for the sum of interior angles
Now, we can perform the arithmetic operations to find the sum:
Sum of interior angles = (3) × 180° = 540°
Thus, the sum of the interior angles in a pentagon is 540°.
Finding the Measure of Each Angle in a Regular Pentagon
Step 1: Determine that the pentagon is regular
To find the measure of each angle in a pentagon, we need to know if the pentagon is regular. A regular pentagon has equal side lengths and equal angle measures, making our calculations simpler.
Step 2: Divide the sum of interior angles by the number of sides
Since all angles are equal in a regular pentagon, we can find the measure of each angle by dividing the sum of interior angles (calculated previously) by the number of sides:
Measure of each angle = Sum of interior angles ÷ Number of sides
Step 3: Calculate the measure of each angle
Using the values we found for a pentagon:
Measure of each angle = 540° ÷ 5 = 108°
Thus, each angle in a regular pentagon measures 108°.
Example – Solving for Angles in an Irregular Pentagon
Suppose we have an irregular pentagon with four known angles: 100°, 110°, 120°, and 130°. To find the missing angle, we can follow these steps:
Step 1: Recall that the sum of interior angles in a pentagon is 540°
From our previous calculations, we know that the sum of interior angles in any pentagon is 540°.
Step 2: Calculate the sum of the known angles
Add the four given angles together:
100° + 110° + 120° + 130° = 460°
Step 3: Find the missing angle
Subtract the sum of the known angles from the total sum of interior angles:
Missing angle = 540° – 460° = 80°
Thus, the missing angle in this irregular pentagon is 80°.
In conclusion, understanding angles in a pentagon involves knowing the polygon angle-sum formula, recognizing if the pentagon is regular or irregular, and applying the appropriate calculations. By following these step-by-step instructions, you can now confidently calculate the angles in any pentagon.