In A Regular Hexagon With Side Length Of 40cm, Find The Length Of The Apothem?


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1: Draw your hexagon, so you can see what is going on.
2: Construct lines from the center point to each of the vertices, you now have 6 isosceles triangles.
3: Divide 360' by 6 to get the angle at the top vertex of one of the triangles = 60'
4: Since these are isosceles triangles, you know the base angles are equal, so these are both 60'

5: This tells you that you are in fact dealing with equilateral triangles not isosceles, so the all three vertices are 40cm

6: Now draw a diagram of an equilateral triangle with three angles of 60' and lengths of 40cm
7: Drop a perpendicular from the vertex to the opposite side - this is your Apothem.  This perpendicular actually creates two right angled triangles, and bisects the side.  So now you have another triangle - yip you guessed it - Draw a diagram.

8: Draw a diagram of a right angled triangle, with hypotenuse of 40, angles of 30, 60 and 90 and shortest length of 20

9: Let A (for Apothem) be the unknown side - use pythagorean theorem to solve.

10 40^2 = 20^2 + A^2

1600 = 400 + A^2
1200 = A^2

square root of 1200 = A  approximately 34.6414cm

I hope this helps


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