Regular Hexagons

Regular Polygons

A polygon is a closed figure made of straight sides. A regular polygon has equal side lengths and equal interior angles. An equilateral triangle is a regular three-sided polygon, and a square is a regular four-sided polygon. Equal sides alone do not make every polygon regular.

The source asks you to explore drawing pentagons and hexagons with equal sides. The exact regular-pentagon construction requires ideas developed later; we will not assume it has already been learned. A regular hexagon, however, can be built using equilateral triangles and the angle constructions available now.

A useful approach is to start with pieces we know how to construct and ask whether they form the required whole. This is stronger than drawing a six-sided figure and deciding by appearance that it must be regular.

Regular Hexagon and Equilateral Triangles

Imagine six congruent equilateral triangles with one vertex of each at a common centre O. Each triangle has three equal sides and three angles of 60°. We need to know whether six such pieces fit around O without leaving a gap or overlapping.

The angles around O total 6 × 60° = 360°, so the pieces exactly fill the turn. Each outer side is one side of an equal triangle, so the six outer sides match. At each outer vertex, two 60° triangle angles meet; the interior angle is 120°. Thus both requirements for regularity are satisfied.

Joining opposite vertices through O gives a straight line because three successive central angles total 3 × 60° = 180°. This explains the opposite-vertex lines in the diagram. It also shows that the distance from O to any vertex equals an outer side length: both are sides of the same equilateral pieces.

Angles Around a Point

A full turn gives a way to check whether pieces can fit at one meeting point. Their angles must add to 360° to cover that point without gaps or overlaps. A small missing angle is still a gap; an excess means some pieces would overlap.

Construction of a 60° Angle

An equilateral triangle supplies an exact 60° angle. We can produce that triangle using equal compass distances. This method turns a length equality into the angle we need.

1. Start with ray AX. With centre A and a chosen radius, mark B on the ray.
2. With the same radius and centre B, draw an arc meeting the first circle or arc at C on the chosen side.
3. Draw AC. AB, BC, and CA are all equal to the compass radius.
4. Triangle ABC is equilateral, so ∠CAB = ∠CAX = 60°.

The adjacent angle on the straight line is 180° − 60° = 120°. You can therefore obtain the interior angle needed for a regular hexagon from the same construction. Keep track of which ray gives 60° and which straight-line continuation gives its 120° neighbour.

Constructing a Regular Hexagon

The six-triangle view provides a direct compass construction. Because the radius from the centre to a vertex equals the side length, the same compass opening can draw the circle and step around its circumference. Each step builds another equilateral triangle.

1. Choose the required side length s and draw a circle with centre O and radius s.
2. Mark one point A on the circle. Keep the compass opening equal to s.
3. From A, mark the next circle point B. Continue in the same direction to C, D, E, and F.
4. The next step returns to A because six 60° central angles make 360°. Join consecutive vertices to obtain the hexagon.
5. Keep the circle and centre visible until you have checked the six equal sides and triangle arrangement.

Check Your Understanding

Use these questions to check the conditions behind each method, as well as the result. For a diagram-based question, follow the labels and given information rather than judging by appearance.