Chek L.

# 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

or

(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.
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06/02/18

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