GoalCut a convex polygon into triangles from one vertex to get the interior angle sum (n − 2) × 180°, then find missing angles and the number of sides.
New words
polygon · a closed shape made of straight sides, named by its number of sidesconvex · no inside angle is more than 180°; the shape has no dentdiagonal · a segment joining two vertices that are not next to each otherinterior angles · the angles inside the polygon at its vertices
Explanation
Pick one vertex of a convex polygon and draw all the diagonals from it. They cut the polygon into triangles. A pentagon gives 3 triangles, a hexagon 4: in general an n-sided polygon gives n − 2 triangles. Each triangle has an angle sum of 180°, and the triangles’ angles together fill exactly the polygon’s interior angles. So the interior angle sum is (n − 2) × 180°. Check: a triangle has 180°, a quadrilateral 360°, a pentagon 540° and a hexagon 720°. Be careful with n − 2, not n. The formula is for convex polygons; a polygon with a dent is not treated here. To find the number of sides from a given angle sum, solve the equation: (n − 2) × 180 = 1440 gives n − 2 = 8, so n = 10. To find a missing angle, subtract the known angles from the total.
Worked examples
Find the interior angle sum of a hexagon. A hexagon has n = 6, so 6 − 2 = 4 triangles and the sum is 4 × 180° = 720°. A decagon (10 sides) has (10 − 2) × 180° = 1440°.
The interior angle sum of a polygon is 1440°. How many sides? (n − 2) × 180 = 1440, so n − 2 = 1440 ÷ 180 = 8 and n = 10. A polygon cannot have a sum of 1000° because 1000 ÷ 180 is not a whole number.
Four angles of a pentagon are 100°, 110°, 120° and 95°. The sum is 540°, so the fifth angle is x = 540° − (100° + 110° + 120° + 95°) = 540° − 425° = 115°.
Class activity
Cut it up. Draw a convex hexagon on paper and draw all the diagonals from one vertex. Count the triangles. Then cut off the angles of a paper hexagon and place them around a point: do they add up to 720°?
Practice
1
Find the interior angle sum of an octagon (8 sides).
(8 − 2) × 180° = 1080°.
2
The interior angle sum is 2340°. How many sides?
2340 ÷ 180 = 13, so n = 15.
3
Three angles of a quadrilateral are 85°, 100° and 90°. Find the fourth.
360° − 275° = 85°.
4
The angles of a hexagon are x, x + 10°, x + 20°, x + 30°, x + 40° and x + 50°. Find x.
6x + 150° = 720°, so x = 95°.
5
Noor says the angle sum of a pentagon is 5 × 180° = 900°. What went wrong?
She used n instead of n − 2. The sum is 3 × 180° = 540°.
