How would you find the area of an equilateral triangle using the Pythagorean Theorem?

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How would you find the area of an equilateral triangle using the Pythagorean Theorem?

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Answer 1 · 2015-12-01T03:09:12

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We can see that if we split an equilateral triangle in half, we are left with two congruent equilateral triangles. Thus, one of the legs of the triangle is $\frac{1}{2} s$ $1/2s$ , and the hypotenuse is $s$ $s$ .

If we want to find the height, we use the Pythagorean Theorem:

${(\frac{1}{2} s)}^{2} + h^{2} = {(s)}^{2}$ $(1/2s)^2+h^2=(s)^2$
$\frac{1}{4} s^{2} + h^{2} = s^{2}$ $1/4s^2+h^2=s^2$
$h^{2} = \frac{3}{4} s^{2}$ $h^2=3/4s^2$
$h = \frac{\sqrt{3}}{2} s$ $h=sqrt3/2s$

If we want to determine the area of the entire triangle, we know that $A = \frac{1}{2} b h$ $A=1/2bh$ . We also know that the base is $s$ $s$ and the height is $\frac{\sqrt{3}}{2} s$ $sqrt3/2s$ , so we can plug those in to the area equation to see the following for an equilateral triangle:

$A = \frac{1}{2} b h \Rightarrow \frac{1}{2} (s) (\frac{\sqrt{3}}{2} s) = \frac{s^{2} \sqrt{3}}{4}$ $A=1/2bh=>1/2(s)(sqrt3/2s)=(s^2sqrt3)/4$

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How would you find the area of an equilateral triangle using the Pythagorean Theorem?

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