Rectilinear Figures [Quadrilaterals: Parallelogram, Rectangle, Rhombus, Square and Trapezium]
237. The Angle Sum Secret · Why every n-gon hides (2n-4) right angles
Drag the vertices to reshape the hexagon. Notice that the sum of interior angles always stays at 720°, which equals 8 right angles.
The interior angles of a polygon with n sides add up to (n − 2) × 180°, the same as (2n − 4) right angles. The polygon splits into (n − 2) triangles, each giving 180°, so a hexagon (n = 6) totals 720°.
What this lesson covers
Try to break it
Drag any vertex of the hexagon. The six interior angles all shift, but their sum stays locked at (6 − 2) × 180° = 720°. Try to drag the vertices to a configuration where the total drifts off 720° — impossible.
How you build it
Make a hexagon.
- Place point A as the first corner of the hexagon.
- Place point B as the next corner.
- Place point C as the next corner.
- Place point D as the next corner.
- Place point E as the next corner.
- Place point F as the last corner.
- Draw segment AB.
- Draw segment BC.
- Draw segment CD.
- Draw segment DE.
- Draw segment EF.
- Draw segment FA to close the hexagon.
The proof, step by step
Prove that the interior angles of a hexagon add up to 720°.
- Pick a vertex, say A.
- Draw diagonals from A to all non-adjacent vertices (C, D, E).
- These diagonals divide the hexagon into 4 triangles.
- Sum of angles in 4 triangles = 4 × 180° = 720°.
- Generalizing: an n-gon splits into (n-2) triangles, so sum = (n-2) × 180° = (2n-4) right angles.
Worked example
The sum of interior angles of a polygon is 1440°. How many sides does the polygon have?
Sum = (n-2)×180°. 1440 = (n-2)×180 ⇒ n-2 = 8 ⇒ n = 10.
- 8
- 9
- 10 — correct
- 12