163. The Polygon's Angle Secret · how diagonals unlock the sum of interior angles
The sum of interior angles of the pentagon equals 540° (3×180°). The sum of interior angles of an n-sided polygon is (n − 2) × 180°.
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For subscribers · Selina ICSE: UNDERSTANDING SHAPES (Including Polygons)
What this lesson covers
Try to break it
Drag any pentagon vertex. The diagonals from A always split the pentagon into 3 triangles, and the interior angles always sum to 3 × 180° = 540°. Try to drag a vertex so the sum lands on anything else — impossible.
How you build it
Construct a hexagon with diagonals from one vertex.
- Draw a polygon with 6 sides.
- Select one vertex and draw diagonals to all non-adjacent vertices.
The proof, step by step
Prove that the interior angles of a pentagon add up to 540°.
- Consider a polygon with n sides.
- Choose any one vertex and draw all possible diagonals from it.
- These diagonals divide the polygon into exactly (n - 2) triangles.
- We know that the sum of interior angles of a triangle is 180°.
- Therefore, the sum of interior angles of the polygon = (n - 2) × 180°.
Worked example
The sum of the interior angles of a polygon is 1080°. Find the number of sides of the polygon.
Let the number of sides be n. Sum = (n - 2) × 180°. So, (n - 2) × 180 = 1080 ⇒ n - 2 = 6 ⇒ n = 8. The polygon has 8 sides.
- 6
- 7
- 8 — correct
- 9