You can find the measure of an interior angle of a polygon by subtracting its corresponding exterior angle from 180 degrees.
The relationship between an interior angle and its adjacent exterior angle at any vertex of a polygon is that they form a linear pair. This means they lie on a straight line when one side of the polygon is extended, and therefore, their measures add up to 180°.
The Core Formula
The fundamental way to calculate an interior angle using an exterior angle is:
Interior Angle = 180° – Exterior Angle
This formula works because an interior angle and its related exterior angle at the same vertex are supplementary, always summing to 180°.
Applying the Formula: An Example
Let's consider the scenario mentioned in the reference:
Suppose a regular polygon has an exterior angle measuring 72°.
To find the size of each interior angle in this regular polygon, we apply the formula:
- Subtract the exterior angle (72°) from 180°.
- Calculation: 180° – 72° = 108°
Therefore, each interior angle in this regular polygon measures 108°.
This method is straightforward and applies to any vertex of any polygon (regular or irregular), as the interior and exterior angles at a single vertex always form a linear pair.
Quick Reference Table
Here's a simple way to visualize the relationship using the example:
| Angle Type | Measure | Calculation |
|---|---|---|
| Exterior Angle | 72° | (Given) |
| Interior Angle | 108° | 180° – 72° = 108° |
Key Takeaway
Remember that the sum of an interior angle and its adjacent exterior angle at the same vertex is always 180°. This supplementary relationship is the key to finding one when you know the other.
This method is particularly useful for regular polygons where all exterior angles are equal, making it easy to find the measure of all interior angles once one exterior angle is known.
