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## 25.2 Regular polygons

Regular polygons are polygons that have all sides equal and all interior angles equal. Both of these conditions must hold true for a polygon to be classified as a regular polygon. If a polygon has all sides equal, that does not necessarily mean that it is a regular shape. For example, a rhombus is a quadrilateral with all sides equal in length, but its interior angles are not equal. A square is a regular polygon because all its sides are equal and all its interior angles are equal (right angles).

An equilateral triangle has $$3$$ equal sides and all interior angles equal $$60^{\circ}$$.

A square has $$4$$ equal sides and all interior angles equal $$90^{\circ}$$.

A regular pentagon has $$5$$ equal sides and all interior angles equal $$108^{\circ}$$.

A regular hexagon has $$6$$ equal sides and all interior angles equal $$120^{\circ}$$.

A regular heptagon has $$7$$ equal sides and all interior angles equal $$\text{128,57}^{\circ}$$.

A regular octagon has $$8$$ equal sides and all interior angles equal $$135^{\circ}$$.

A regular nonagon has $$9$$ equal sides and all interior angles equal $$140^{\circ}$$.

A regular decagon has $$10$$ equal sides and all interior angles equal $$144^{\circ}$$.

In terms of regular polygons, think about the following:

• As the number of sides of a regular polygon increases, what do you notice about the shape of the polygon? What shape is the polygon starting to resemble?
• What do you notice about the size of the interior angles as the number of sides of the regular polygon increases?
• What size of angle do you think these values are tending towards?

## Worked Example 25.1: Drawing a regular pentagon

Use a ruler, compass and protractor to draw a regular pentagon with sides equal to $$5 \text{ cm}$$.

### Draw side $$AB = 5 \text{ cm}$$.

Use a ruler and a compass to draw side $$AB = 5 \text{ cm}$$. At vertex $$A$$, measure an angle of $$108^{\circ}$$ with a protractor and draw a line. At vertex $$B$$, measure an angle of $$108^{\circ}$$ and draw a line.

### Use your compass to measure $$5 \text{ cm}$$.

Use your compass to measure $$5 \text{ cm}$$ on these two lines. Place the point of the compass of $$A$$ and mark off point $$E$$ on the line. In the same way, place the point of the compass of $$B$$ and mark off point $$C$$ on the line.

### Draw the final vertex $$D$$.

At vertex $$E$$ measure an angle of $$108^{\circ}$$ and draw a line. Use your compass to measure $$5 \text{ cm}$$. Place the point of the compass of $$E$$ and mark off point $$D$$ on the line. Draw line $$CD$$ to complete the regular pentagon.

### Construct regular polygons

1. Use a compass and a protractor to draw an equilateral triangle with sides equal to $$6 \text{ cm}$$.
2. Use a compass and a protractor to draw a regular hexagon with sides equal to $$4 \text{ cm}$$.
3. Use a compass and a protractor to draw a cube with sides equal to $$5 \text{ cm}$$.
4. Use a compass and a protractor to draw a regular pentagon with sides equal to $$4 \text{ cm}$$.