Explanation
In this page, we practice finding the sum of the interior angles of different polygons.
Remember:
A polygon can be split into triangles.
Since each triangle has an angle sum of \(180^\circ\), we can use the number of triangles inside the polygon to find the total sum of the interior angles.
For a polygon with \(n\) sides, the number of triangles is:
\(n − 2\)
So, the formula is:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
Formula / Rule
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
where:
\(n\) is the number of sides of the polygon.
Example 1
Find the sum of the interior angles of a quadrilateral.
A quadrilateral has 4 sides, so:
\(n = 4\)
Use the formula:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
Substitute \(n = 4\):
Sum of Interior Angles \(= (4 − 2) × 180^\circ\)
\(= 2 × 180^\circ\)
\(= 360^\circ\)
So, the sum of the interior angles of a quadrilateral is:
\(360^\circ\)
Example 2
Find the sum of the interior angles of a pentagon.
A pentagon has 5 sides, so:
\(n = 5\)
Use the formula:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
Substitute \(n = 5\):
Sum of Interior Angles \(= (5 − 2) × 180^\circ\)
\(= 3 × 180^\circ\)
\(= 540^\circ\)
So, the sum of the interior angles of a pentagon is:
\(540^\circ\)
Example 3
Find the sum of the interior angles of a hexagon.
A hexagon has \(6\) sides, so:
\(n = 6\)
Use the formula:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
Substitute \(n = 6\):
Sum of Interior Angles \(= (6 − 2) × 180^\circ\)
\(= 4 × 180^\circ\)
\(= 720^\circ\)
So, the sum of the interior angles of a hexagon is:
\(720^\circ\)
Example 4
Find the sum of the interior angles of a \(1002\)–sided polygon.
A \(1002\)–sided polygon has \(1002\) sides, so:
\(n = 1002\)
Use the formula:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
Substitute \(n = 1002\):
Sum of Interior Angles \(= (1002 − 2) × 180^\circ\)
\(= 1000 × 180^\circ\)
\(= 180,000^\circ\)
So, the sum of the interior angles of a \(1002\)–sided polygon is:
\(180,000^\circ\)
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?