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Chek L.

asked • 06/02/18

Area of largesz equilateral triangle inside square

What is the area of the largest equilateral triangle which fits inside a square of side a? 

Mark O.

Hi Chek,
I have done the problem a second way. I know the area of the square is a2. A triangle of base b nd height h has an area of (1/2)bh.
But, (1/2)bh ≤ a2
If we have an equilateral triangle, then we can draw a vertical from the upper vertex as I did below. You would have a right triangle of base b/2, height h and hypotenuse b, which agrees with the base length b since the overall triangle is equilateral. Then, using the Pythagorean theorem, h = (b/2)√3
Therefore, the above inequality becomes
(1/2)b((b/2)√3) ≤ a2
(b2 / 4)√3 ≤ a2
(b/2)(31/4)  ≤ a
b ≤ (2a)/(31/4)
So Amax = (1/2)bh = (1/2)b(b/2)√3 = (b2/4)√3 = (1/4)((2a)/(31/4))2√3 = a2
Now I get the maximum area of the equilateral triangle is the same as the square, a2.


2 Answers By Expert Tutors


Mark O. answered • 06/02/18

5.0 (167)

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