The sum of the interior angles of an n-gon is 180°×(n−2). This can be derived by dividing the polygon into triangles.
Example ①
Find the sum of the interior angles of a hexagon.
Answer: 180°×(6−2) = 180°×4 = 720°
Find the sum of the interior angles of a hexagon.
Answer: 180°×(6−2) = 180°×4 = 720°
Example ②
A regular polygon has one interior angle measuring 144°. Find the number of its sides.
Answer: One interior angle = 180°(n−2)/n = 144°. 180n−360=144n → 36n=360 → n=10. Regular decagon
A regular polygon has one interior angle measuring 144°. Find the number of its sides.
Answer: One interior angle = 180°(n−2)/n = 144°. 180n−360=144n → 36n=360 → n=10. Regular decagon
Key Points
- Sum of the interior angles of an n-gon = 180°×(n−2)
- One interior angle of a regular n-gon = 180°(n−2)/n
- When dividing into triangles, fix one vertex to form (n−2) triangles
Practice Problems
- Find the sum of the interior angles of an octagon.
- What polygon has an interior angle sum of 1260°?
Answer and Explanation
- Answer: 1080°. 180×(8−2)=1080.
- Answer: Nonagon. 180(n−2)=1260→n−2=7→n=9.