Asked by Liana
Why is the apothem of an equilateral triangle one third its height?
Answers
Answered by
MathMate
The apothem of a regular polygon is the distance from the centre to one of its sides.
Let the equilateral triangle (regular polygon of three sides) be ABC, centred at O. Drop a perpendicular from O to D on side AB.
Consider triangle OAB, ∠OAB and ∠OBA are each 30°.
Since sin(30°)=1/2, it is clear that OD = OA/2 = OC/2
Therefore
Apothem OD=(OD+OA)/3=CD/3=height/3.
Let the equilateral triangle (regular polygon of three sides) be ABC, centred at O. Drop a perpendicular from O to D on side AB.
Consider triangle OAB, ∠OAB and ∠OBA are each 30°.
Since sin(30°)=1/2, it is clear that OD = OA/2 = OC/2
Therefore
Apothem OD=(OD+OA)/3=CD/3=height/3.
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