A Regular Hexagon = Six Equilateral Triangles
Area of a regular hexagon? Draw lines from the center to each vertex and it splits into six equilateral triangles — related to why bees build hexagons.
Problem
Find the area of a regular hexagon (side s) by splitting it from the center into equilateral triangles.
Cut · rotate · rearrange
Press Play or move Next one step at a time. Follow the pieces until the answer becomes visible. (Arrow keys and Space also work.)
A regular hexagon (side s). How to find its area?
Why this approach works
A regular hexagon has radius equal to its side. Lines from the center make six triangles, each with two sides equal to the radius (= s) and an included angle of 360°/6 = 60° — so each is equilateral. Thus the hexagon is six equilateral triangles: 6 × (√3/4)s² = (3√3/2)s² ≈ 2.598 s². Splitting into known shapes is the core strategy for polygon area.
Proof
Since a regular hexagon has radius equal to its side, the six central triangles are equilateral (two sides s, 60° between). Six times (√3/4)s² gives the hexagon’s area.
Try it yourself
Any regular n-gon splits into n isosceles triangles; as n grows it nears a circle and the area tends to πr² (see #9).
Related mathematics
Beyond MathVoyage
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