Explanation
In this lesson, we learn how to find missing interior angles of a polygon.
The interior angles of a polygon are the angles inside the shape.
To find a missing interior angle, we first need to know the sum of the interior angles of the polygon.
For a polygon with \(n\) sides, the sum of the interior angles is:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
where:
\(n\) is the number of sides of the polygon.
There are two common cases:
Regular Polygons
A regular polygon has:
• all sides equal
• all interior angles equal
So, after finding the sum of the interior angles, we divide by the number of angles.
Irregular Polygons
An irregular polygon does not have all sides or all angles equal.
To find a missing angle in an irregular polygon, we:
- Find the sum of the interior angles.
- Add the known angles.
- Subtract the known angles from the total sum.
Formula / Rule
For any polygon:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
For a regular polygon:
Each Interior Angle \(=\frac{\text{Sum of Interior Angles}}{n}\)
or:
Each Interior Angle \(= \frac{(n − 2) × 180^\circ}{n}\)
For an irregular polygon:
Missing Angle = Sum of Interior Angles − Sum of Known Angles
Examples
Example 1: Regular Polygon
Find each interior angle of a regular pentagon.
A pentagon has \(5\) sides, so:
\(n = 5\)
First, find the sum of the interior angles:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
\(= (5 − 2) × 180^\circ\)
\(= 3 × 180^\circ\)
\(= 540^\circ\)
Since the pentagon is regular, all 5 angles are equal.
So:
Each Interior Angle = \frac{540^\circ}{5}
\(= 108^\circ\)
So, each interior angle of a regular pentagon is:
\(108^\circ\)
Example 2: Irregular Polygon
Find the missing angle in a quadrilateral with angles:
\(50^\circ, 130^\circ, 50^\circ,\) and \(x\)
A quadrilateral has \(4\) sides, so:
\(n = 4\)
First, find the sum of the interior angles:
Sum of Interior Angles \(= (4 − 2) × 180^\circ\)
\(= 2 × 180^\circ\)
\(= 360^\circ\)
Now subtract the known angles:
\(x = 360^\circ − (50^\circ + 130^\circ + 50^\circ)\)
\(x = 360^\circ − 230^\circ\)
\(x = 130^\circ\)
So, the missing angle is:
\(130^\circ\)
Video Explanation
Practice
Continue Learning
- Sum of Interior Angles of a Polygon (Proof)
- Find Missing Interior Angles of a Polygon (Examples) (Current lesson)
- Why Regular and Irregular Polygons Have Different Angle Formulas?