Explanation
In this page, we practice why we divide for regular polygons and why we subtract for irregular polygons.
Remember:
For any polygon:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
If the polygon is regular, all angles are equal, so we divide the sum by the number of angles.
If the polygon is irregular, the angles are not all equal, so we subtract the given angles from the total sum.
Formula / Rule
For a regular polygon:
\(x = \frac{\text{Sum}}{n}\)
For an irregular polygon:
\(x =\) Sum \(−\) Sum of Known Angles
Example 1
Find each interior angle of a regular triangle.
A regular triangle has \(3\) equal angles.
So:
Sum \(= x + x + x\)
Sum \(= 3x\)
Divide both sides by \(3\):
\(x = \frac{\text{Sum}}{3}\)
A triangle has an interior angle sum of:
\(180^\circ\)
So:
\(x = \frac{180^\circ}{3}\)
\(x = 60^\circ\)
So, each interior angle is:
\(60^\circ\)
Example 2
Find each interior angle of a regular quadrilateral.
A regular quadrilateral has \(4\) equal angles.
So:
Sum \(= x + x + x + x\)
Sum \(= 4x\)
Divide both sides by \(4\):
\(x = \frac{\text{Sum}}{4}\)
A quadrilateral has an interior angle sum of:
\(360^\circ\)
So:
\(x = \frac{360^\circ}{4}\)
\(x = 90^\circ\)
So, each interior angle is:
\(90^\circ\)
Example 3
Find the missing angle in an irregular triangle with angles:
\(90^\circ, 40^\circ,\) and \(x\)
The sum of the interior angles of a triangle is:
\(180^\circ\)
So:
Sum \(= 90^\circ + 40^\circ + x\)
Subtract the known angles:
\(x = 180^\circ − (90^\circ + 40^\circ)\)
\(x = 180° − 130^\circ\)
\(x = 50^\circ\)
So, the missing angle is:
\(50^\circ\)
Example 4
Find the missing angle in an irregular quadrilateral with angles:
\(50^\circ, 50^\circ, 130^\circ,\) and \(x\)
The sum of the interior angles of a quadrilateral is:
\(360^\circ\)
So:
Sum \(= 50^\circ + 50^\circ + 130^\circ + x\)
Subtract the known angles:
\(x = 360^\circ − (50^\circ + 50^\circ + 130^\circ)\)
\(x = 360^\circ − 230^\circ\)
\(x = 130^\circ\)
So, the missing angle is:
\(130^\circ\)
Example 5
Find each interior angle of a regular pentagon.
A pentagon has \(5\) sides, so:
\(n = 5\)
First, find the sum:
Sum \(= (5 − 2) × 180^\circ\)
Sum \(= 3 × 180^\circ\)
Sum \(= 540^\circ\)
Since the pentagon is regular, all \(5\) angles are equal.
So:
\(x = \frac{540^\circ}{5}\)
\(x = 108^\circ\)
So, each interior angle is:
\(108^\circ\)
Example 6
Find the missing angle in an irregular pentagon with angles:
\(90^\circ, 90^\circ, 130^\circ, 130^\circ,\) and \(x\)
A pentagon has \(5\) sides, so:
\(n = 5\)
First, find the sum:
Sum \(= (5 − 2) × 180^\circ\)
Sum \(= 3 × 180^\circ\)
Sum \(= 540^\circ\)
Now subtract the known angles:
\(x = 540^\circ − (90^\circ + 90^\circ + 130^\circ + 130^\circ)\)
\(x = 540^\circ − 440^\circ\)
\(x = 100^\circ\)
So, the missing angle is:
\(100^\circ\)
Practice
- More Examples (Current page)
Continue Learning
- Sum of Interior Angles of a Polygon (Proof)
- Find Missing Interior Angles of a Polygon (Examples)
- Why Regular and Irregular Polygons Have Different Angle Formulas?