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
- Use a compass and a protractor to draw an equilateral triangle with sides equal to \(6 \text{ cm}\).
- Use a compass and a protractor to draw a regular hexagon with sides equal to \(4 \text{ cm}\).
- Use a compass and a protractor to draw a cube with sides equal to \(5 \text{ cm}\).
- Use a compass and a protractor to draw a regular pentagon with sides equal to \(4 \text{ cm}\).