Explanation
In this page, we practice finding missing interior angles in both regular and irregular polygons.
Remember:
For any polygon:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
For a regular polygon, all interior angles are equal, so we divide the total sum by the number of angles.
For an irregular polygon, we 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 \(= \frac{\text{Sum of Interior Angles}}{n}\)
For an irregular polygon:
Missing Angle \(=\) Sum of Interior Angles \(−\) Sum of Known Angles
Example 1
Find each interior angle of a regular triangle.
A regular triangle has \(3\) equal angles.
A triangle has \(3\) sides, so:
\(n = 3\)
First, find the sum:
Sum of Interior Angles \(= (3 − 2) × 180^\circ\)
\(= 1 × 180^\circ\)
\(= 180^\circ\)
Now divide by \(3\):
Each Interior Angle \(= \frac{180^\circ}{3}\)
\(= 60^\circ\)
So, each interior angle of a regular triangle is:
\(60^\circ\)
Example 2
Find each interior angle of a regular quadrilateral.
A regular quadrilateral has \(4\) equal angles.
A quadrilateral has \(4\) sides, so:
\(n = 4\)
First, find the sum:
Sum of Interior Angles \(= (4 − 2) × 180^\circ\)
\(= 2 × 180^\circ\)
\(= 360^\circ\)
Now divide by \(4\):
Each Interior Angle \(= \frac{360^\circ}{4}\)
\(= 90^\circ\)
So, each interior angle of a regular quadrilateral is:
\(90^\circ\)
This regular quadrilateral is a square.
Example 3
Find each interior angle of a regular hexagon.
A hexagon has 6 sides, so:
\(n = 6\)
First, find the sum:
Sum of Interior Angles \(= (6 − 2) × 180^\circ\)
\(= 4 × 180^\circ\)
\(= 720^\circ\)
Since the hexagon is regular, all 6 angles are equal.
So:
Each Interior Angle \(= \frac{720^\circ}{6}\)
\(= 120^\circ\)
So, each interior angle of a regular hexagon is:
\(120^\circ\)
Example 4
Find the missing angle in a triangle with angles:
\(40^\circ, 90^\circ,\) and \(x\)
A triangle has \(3\) sides, so the sum of the interior angles is:
\(180^\circ\)
Now subtract the known angles:
\(x = 180^\circ − (40^\circ + 90^\circ)\)
\(x = 180^\circ − 130^\circ\)
\(x = 50^\circ\)
So, the missing angle is:
\(50^\circ\)
Example 5
Find the missing angle in a quadrilateral with angles:
\(50^\circ, x, 130^\circ,\) and \(50^\circ\)
A quadrilateral has \(4\) sides, so:
\(n = 4\)
Find the sum:
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:
\(130^\circ\)
Example 6
Find the missing angle in a pentagon with angles:
\(90^\circ, 90^\circ, 130^\circ, x,\) and \(130^\circ\)
A pentagon has \(5\) sides, so:
\(n = 5\)
Find the sum:
Sum of Interior Angles \(= (5 − 2) × 180^\circ\)
\(= 3 × 180^\circ\)
\(= 540^\circ\)
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:
\(100^\circ\)
Example 7
Find the missing angle in a hexagon with angles:
\(90^\circ, 120^\circ, x, 40^\circ, 270^\circ,\) and \(90^\circ\)
A hexagon has \(6\) sides, so:
\(n = 6\)
Find the sum:
Sum of Interior Angles \(= (6 − 2) × 180^\circ\)
\(= 4 × 180^\circ\)
\(= 720^\circ\)
Now subtract the known angles:
\(x = 720^\circ − (90^\circ + 120^\circ + 40^\circ + 270^\circ + 90^\circ)\)
\(x = 720^\circ − 610^\circ\)
\(x = 110^\circ\)
So, the missing angle is:
\(110^\circ\)
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?