Exercise solution 2, page 172 - GCSE Mathematics for OCR Higher Student Book

⭐ What you need to know:

A) A regular polygon is a polygon in which all sides are equal and all interior angles are equal. This means that if you know one interior angle, you know them all.

B) It is important to distinguish between an interior angle and an exterior angle.

The interior angle is the angle inside the polygon. The exterior angle is formed outside the polygon when one side is extended.

For a regular polygon, each interior angle and its corresponding exterior angle add up to \(180^\circ\), because they form a straight line:

\[ \text{interior angle} + \text{exterior angle} = 180^\circ \]

For example, if an interior angle is \(150^\circ\), then the exterior angle is:

\[\begin{aligned} \text{exterior angle} &= 180^\circ - 150^\circ \\ &= 30^\circ \end{aligned}\]

C) In any regular polygon, the sum of the exterior angles is always \(360^\circ\).

This is very useful because once you know one exterior angle, you can find the number of sides using:

\[ \text{number of sides} = \frac{360^\circ}{\text{exterior angle}} \]

For example, if each exterior angle is \(45^\circ\), then:

\[\begin{aligned} \text{number of sides} &= \frac{360}{45} \\ &= 8 \end{aligned}\]

So the polygon would have 8 sides.

D) Once you know the number of sides, you can identify the name of the polygon.

Some common examples are:

3 sides: triangle
4 sides: quadrilateral
5 sides: pentagon
6 sides: hexagon
7 sides: heptagon
8 sides: octagon
9 sides: nonagon
10 sides: decagon

E) A good method for this type of question is:

1. Use \(180^\circ - \text{interior angle}\) to find the exterior angle.

2. Use \(\dfrac{360^\circ}{\text{exterior angle}}\) to find the number of sides.

3. Name the polygon from its number of sides.

This step-by-step approach helps avoid confusion and makes the problem easier to solve.