Numerical answer
Each interior angle of a regular polygon is 140°. How many diagonals does the polygon have?
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Answer: 27
Exterior angle = 180° − 140° = 40°.
Number of sides n = 360°/40° = 9.
Number of diagonals = n(n − 3)/2 = 9 × 6/2 = 27.
Why n(n − 3)/2: each vertex joins to n − 3 others by a diagonal (not itself, not its two neighbours), and each diagonal is counted twice.
Tip: Work with the exterior angle; it gives n directly.
Number of sides n = 360°/40° = 9.
Number of diagonals = n(n − 3)/2 = 9 × 6/2 = 27.
Why n(n − 3)/2: each vertex joins to n − 3 others by a diagonal (not itself, not its two neighbours), and each diagonal is counted twice.
Tip: Work with the exterior angle; it gives n directly.
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