Regular hexagon from equilateral triangles

The idea

Since six 60° angles add up to 360°, six congruent equilateral triangles fit around a point without gaps or overlaps, forming a regular hexagon (equal sides, equal angles) whose angles are each 120°.

Triangles round a point

Each angle of an equilateral triangle is 60°. Put the triangles side by side with one corner at the same point O. We want them to fill all the space round O, with no gap and no overlap.

How many triangles do we need?

• 6
• 4
• 8

Add the triangles

Add the triangles one at a time round the point O. Watch the total angle.

A regular hexagon appears

Six angles of 60° add up to 6 × 60° = 360°, so six triangles fit round O. The outside of the shape has 6 sides, and each is a side of an equilateral triangle: all equal. At each corner two 60° angles meet: 60° + 60° = 120°. So it is a regular hexagon.

Six congruent equilateral triangles fit round a point, since 6 × 60° = 360°. They form a regular hexagon: equal sides, and each angle 120°.

• Triangles | Total angle | Result
• 5 | 300° | gap of 60°
• 6 | 360° | fits exactly
• 7 | 420° | overlap

Notes

Six congruent equilateral triangles fit round a point, since 6 × 60° = 360°. They form a regular hexagon: equal sides, and each angle 120°.

Check yourself

A regular hexagon is made from 6 equilateral triangles with side 5 cm. What is the perimeter of the hexagon, in cm?

Answer: 30

The outside has 6 sides of 5 cm each: 6 × 5 = 30 cm.

Why can five equilateral triangles not make a regular hexagon round a point?

At each corner of the regular hexagon two 60° angles meet. How many degrees do the six corner angles of the hexagon add up to?

Answer: 720

Each corner is 60° + 60° = 120°, and 6 × 120° = 720°.

• 5 × 60° = 300°, so a gap of 60° is left. — correct. Yes! A full turn is 360°.
• Because 5 × 60° = 360°, which is too much.. 5 × 60° = 300°, not 360°. It is too little.
• Because equilateral triangles have 90° angles.. They have 60° angles.