Polygons and Regular Polygons

1What is a polygon?

A polygon is a closed, flat shape made only of straight sides. A shape is a polygon when it passes three tests:

• every side is a straight line segment, with no curves;
• it is closed: each side joins exactly one other side at each end, so there are no gaps;
• the sides never cross each other.

The corner where two sides meet is a vertex (plural: vertices). A polygon always has the same number of vertices as sides.

Not polygons: a circle (its edge is curved), the letter C (it is not closed), and a figure of eight drawn with straight lines that cross in the middle.

2Naming polygons

3Regular polygons

A polygon is regular when both of these are true: all its sides are equal, and all its interior angles are equal. One without the other is not enough.

• A rhombus (a diamond, like the ones in imigongo) has four equal sides, but two of its angles are wider than the other two. Not regular.
• A rectangle has four equal angles of 90°, but its long sides are longer than its short sides. Not regular.
• A square has four equal sides and four equal angles of 90°. Regular.
• An equilateral triangle has three equal sides and three equal angles of 60°. Regular.

When a polygon is regular we usually say so in its name: a regular pentagon, a regular hexagon, and so on.

Worked example 1: is this hexagon regular?

Mugisha measures a hexagon. All six sides are 4 cm, and all six angles are 120°.

1. Are all the sides equal? Yes, they are all 4 cm.
2. Are all the angles equal? Yes, they are all 120°.
3. Both tests pass, so it is a regular hexagon.

Worked example 2: is this quadrilateral regular?

Keza measures a quadrilateral. All four sides are 5 cm. Its angles are 70°, 110°, 70° and 110°.

1. Are all the sides equal? Yes.
2. Are all the angles equal? No, because 70° is not the same as 110°.
3. One test fails, so it is not regular. It is a rhombus.

Try it yourself

1. Is a circle a polygon? Explain why.
2. A polygon has 8 vertices. How many sides does it have, and what is its name?
3. A triangle has sides of 5 cm, 5 cm and 5 cm and angles of 60°, 60° and 60°. Is it regular?
4. A rectangle is 6 cm long and 6 cm wide. Is it regular? What is its special name?

Answers

1. No. Its edge is curved, and a polygon has only straight sides.
2. 8 sides. It is an octagon.
3. Yes. Its sides are equal and its angles are equal, so it is a regular (equilateral) triangle.
4. Yes. All four sides are 6 cm and all four angles are 90°, so it is a square, which is a regular polygon.