3
Sum of interior and exterior angles of a convex polygon
Textbook: pp. 11–13
GoalKnow and use the formulas for the sums of the interior and exterior angles of a convex polygon.
New words
interior angle · ichki burchakexterior angle · tashqi burchakregular polygon · muntazam ko‘pburchakangle sum · burchaklar yig‘indisi
Explanation
The sum of the interior angles of a convex n-gon is 180°·(n − 2). To prove it, the diagonals from one vertex cut the polygon into n − 2 triangles, each with angle sum 180°. An exterior angle at a vertex is the angle adjacent to the interior angle there. The exterior angles, one taken at each vertex, add up to 360° whatever the number of sides. A convex polygon with all sides equal and all angles equal is called regular; its interior angle is 180°·(n − 2) : n and its exterior angle is 360° : n.
Worked examples
In a regular 12-gon the interior angle is 180° · 10 : 12 = 150° and the exterior angle is 360° : 12 = 30°; 150° + 30° = 180°.
The interior angles of a convex polygon add up to 1440°. From 180°·(n − 2) = 1440° we get n − 2 = 8, so n = 10.
Class activity
“Split into triangles”: students split a quadrilateral and a pentagon into triangles by diagonals and compare the number of triangles with n − 2.
Practice
1
How many degrees do the interior angles of a convex 15-gon add up to?
2340
2
How many degrees is one interior angle of a regular 9-gon?
140
3
One exterior angle of a regular polygon is 24°. How many sides does it have?
15
4
Why can a convex quadrilateral not have four acute angles?
The angles add up to 360°, but four acute angles would add up to less than 4 · 90° = 360°.