☰ Year plan · Mathematics

Quadrilaterals and polygons

Week 13 · Quadrilaterals and polygons: angle sums, special quadrilaterals, interior angles of polygons · 3 lessons
3

Polygons: convex and concave, diagonals and the sum of the interior angles

Mathematics Prep 1 Term 1, Unit 4, Lessons 4–5
GoalTell convex from concave polygons and use (n − 2) × 180° to find angle sums.
New words
polygon · a closed figure made of straight segmentsdiagonal · a segment joining two vertices that are not next to each otherconvex · every interior angle is less than 180°concave · at least one interior angle is more than 180°
Explanation

A polygon with n sides has n vertices. A diagonal joins two vertices that are not neighbours. From one vertex you can draw n − 3 diagonals, and they cut the polygon into n − 2 triangles. Each triangle has an angle sum of 180°, so the sum of the interior angles of a polygon is (n − 2) × 180°. A polygon is convex when every interior angle is less than 180°, and concave when one angle is a reflex angle, bigger than 180°. In the pentagon below, the dashed diagonals make 3 triangles, and 3 × 180° = 540°.

ABCDE
Worked examples
ABCDEF
Angle sum. A hexagon has n = 6 sides. From one vertex 6 − 3 = 3 diagonals make 6 − 2 = 4 triangles. The sum is 4 × 180° = 720°. Likewise an octagon: (8 − 2) × 180° = 1080°.
A missing angle. Four angles of a pentagon are 100°, 110°, 120° and 95°. They add up to 425°. The pentagon sum is 540°, so the fifth angle is 540° − 425° = 115°.
Find the number of sides. The angles of a polygon add up to 1260°. Then (n − 2) × 180 = 1260, so n − 2 = 7 and n = 9.
ABCD
Convex or concave? In the figure the angle at D points inwards and is more than 180°, so the polygon is concave. A polygon with angles 100°, 90°, 110° and 60° is convex because every angle is less than 180°.
Class activity

Splitting shapes. With a grown-up, draw a heptagon (7 sides) on paper. Pick one vertex and draw all diagonals from it. Count the triangles. Does the number match n − 2?

Practice
1
How many triangles does a heptagon split into from one vertex? What is the sum of its angles?
2
The angles of a polygon add up to 1080°. How many sides has it?
3
Five angles of a hexagon are 120°, 130°, 110°, 125° and 115°. Find the sixth.
4
A polygon has an angle of 200°. Is it convex or concave?
5
Karim says a pentagon splits into 5 triangles, so its angle sum is 5 × 180° = 900°. What is wrong?