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Corners are cut off from an equilateral triangle T to produce a regular hexagon H. What is the ratio of the area of H to the area of T?

 Jun 15, 2020
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Drawing it out would help, but I'm new to the "write code to make funny pictures on web2.0.calc" thing so this is the best I can do.

 

The side of the triangle is ABC = 3a
The side of the hexagon = a
Area of triangle = (root3 /4 )*(3a)^2 = 9a^2 *(root3/4)
Area of hexagon with side a = 6*(root3 /4 )*(a)^2
Ratio of areas of hexagon to that of triangle = { 6*(root3 /4 )*(a)^2 } / { 9a^2 *(root3/4)} = 6/9 = 2/3
and we get = 2 : 3

 

Yay!

 Jun 16, 2020

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