The regular hexagon

A regular polygon has equal sides and equal angles. The regular polygon with 3 sides is the equilateral triangle and the one with 4 sides is the square, and you have constructed both. What about a regular pentagon and a regular hexagon?

The pentagon needs a deeper understanding of triangles and will wait for later years. But the hexagon is within reach — if we break it into pieces we already know how to build.

What happens if you join the OPPOSITE points of a regular hexagon? You get six triangles meeting at the centre. Will they all be equilateral? Rather than answering directly, reverse the approach: can six congruent EQUILATERAL triangles be placed together around a point, and if so, is the result a regular hexagon?

If they can, then the outer sides of the hexagon are all equal (they are triangle sides), and each hexagon angle is made of two triangle angles, 60 + 60 = 120°. So the real question is whether six such triangles fit around the centre WITHOUT overlapping and WITHOUT leaving a gap.

That is a question about angles at a point, and we have the rule for it: a degree is defined by taking the complete angle around a point to be 360°. So the angles around the centre must add to exactly 360°.

Test the idea on a puzzle. Angles of 40°, 60°, 50°, 30°, 40° and 90° are placed around a point, and a gap is left. The gap angle is 360° − (40 + 60 + 50 + 30 + 40 + 90) = 360° − 310° = 50°. So a 70° angle would NOT fit that gap; a 50° one would exactly.

Now the hexagon. Each angle of an equilateral triangle is 60°, and six of them give 6 × 60 = 360° exactly. So six congruent equilateral triangles fit perfectly around the centre — no overlap, no gap — and the result is a regular hexagon. To construct one with side 4 cm, build the six triangles; or construct a 120° angle directly, which you can do because you can construct 60°.