Explanation
The interior angles of a polygon are the angles inside the shape.
A polygon is a closed shape made of straight line segments. Examples of polygons include triangles, quadrilaterals, pentagons, hexagons, and heptagons.
To find the sum of the interior angles of a polygon, we can split the polygon into non-overlapping triangles.
Since the sum of the interior angles of one triangle is:
\(180^\circ\)
we can find the sum of the interior angles of a polygon by counting how many triangles are formed inside it.
For example:
A triangle has \(1\) triangle inside it, so the sum is:
\(1 × 180^\circ = 180^\circ\)
A quadrilateral can be split into \(2\) triangles, so the sum is:
\(2 × 180^\circ = 360^\circ\)
A pentagon can be split into \(3\) triangles, so the sum is:
\(3 × 180^\circ = 540^\circ\)
Each time, the number of triangles is \(2\) less than the number of sides.
So, if a polygon has \(n\) sides, it can be split into:
\((n − 2)\) triangles
This gives us the formula for the sum of the interior angles of any polygon.
Formula / Rule
For a polygon with \(n\) sides:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
where:
\(n\) is the number of sides of the polygon.
This formula gives the total sum of all the interior angles, not just one angle.
Example
Find the sum of the interior angles of a heptagon.
A heptagon has \(7\) sides, so:
\(n = 7\)
Use the formula:
Sum of Interior Angles \(= (n − 2) × 180^\circ\)
Substitute \(n = 7\):
Sum of Interior Angles \(= (7 − 2) × 180^\circ\)
\(= 5 × 180^\circ\)
\(= 900^\circ\)
So, the sum of the interior angles of a heptagon is:
\(900^\circ\)
Video Explanation
Practice
Continue Learning
- Sum of Interior Angles of a Polygon (Proof) (Current lesson)
- Find Missing Interior Angles of a Polygon (Examples)
- Why Regular and Irregular Polygons Have Different Angle Formulas?