Before reading this guide, it may be helpful to read the guides from Module 1 on the properties of angles and the properties of 2D shapes.
Polygons and angles
The sum of the angles inside any polygon can be found by splitting the polygon into triangles.
The number of triangles that a polygon can be divided into is always two less than its number of sides.
| Number of sides | Number of triangles | Sum of angles |
|---|---|---|
| 4 | 2 (4 - 2) | 2x180 =360 |
| 5 | 3 (5 – 2) | 3x180 =540 |
| 6 | 4 (6 – 2) | 4x180 =720 |
| 7 | 5 (7 – 2) | 5x 180 = 900 |
Consider any triangle. The sum of the angles is known to be 180º.
A quadrilateral can be split into two triangles. Each triangle has 180º.
The sum of the angles in a quadrilateral is \(2 \times 180 = 360^{\circ}\).
The sum of the angles in a pentagon is \(3 \times 180 = 540^{\circ}\).
The sum of the angles in a hexagon is \(4 \times 180 = 720^{\circ}\).
The number of triangles is always two less than the number of sides.
Examples
Question
a) Calculate the sum of the angles in this regular heptagon.
b) Calculate the size of each angle.
Solution
a) Split the shape into triangles.
Since there are 5 triangles, the sum of the angles inside the shape is \(5 \times 180 = 900^{\circ}\)
b) The shape is a regular heptagon, so all the angles must be equal.
The total of all 7 equal angles is \(900^{\circ}\)
Each angle \(= 900 \div 7 = 128.6^{\circ}\)
Question
Calculate the value of the missing angle in this shape.
The shape has 5 sides and can be split into 3 triangles.
The sum of the angles is \(3 \times 180 = 540^{\circ}\).
To find the value of \(x\), add together the known angles and subtract the answer from 540°.
\(95 + 95 + 110 + 110 = 410\)
\(540 – 410 = 130^{\circ}\)
\(x = 130^{\circ}\)
Question
Calculate the value of the angle \(y\) in this shape.
The sum of the angles in a quadrilateral is 360°.
\(62 + 90 + 78 = 230\)
\(y = 360 – 230 = 130^{\circ}\)
Test yourself
More on M5: Geometry and measures
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