What is area of equilateral triangle?

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What is area of equilateral triangle?

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Answer 1 · 2015-06-15T18:32:38

If the sides of an equilateral triangle are all of length $a$ $a$ , then the area is $\frac{\sqrt{3}}{4} a^{2}$ $sqrt(3)/4a^2$

Explanation:

Consider an equilateral triangle with sides of length $a$ $a$ .

If you bisect it to make two right angled triangles, then those triangles will have hypotenuse of length $a$ $a$ , shortest side of length $\frac{a}{2}$ $a/2$ and other side of length:

$\sqrt{a^{2} - {(\frac{a}{2})}^{2}} = \sqrt{a^{2} - \frac{a^{2}}{4}} = \sqrt{\frac{3 a^{2}}{4}} = \frac{\sqrt{3} a}{2}$ $sqrt(a^2-(a/2)^2) = sqrt(a^2-a^2/4) = sqrt((3a^2)/4) = (sqrt(3)a)/2$

The two right angled triangles can be rearranged (turning one over) into a rectangle with sides $\frac{\sqrt{3} a}{2}$ $(sqrt(3)a)/2$ and $\frac{a}{2}$ $a/2$ .

The area of the rectangle, which is the same as the area of the original triangle is:

$\frac{\sqrt{3} a}{2} \cdot \frac{a}{2} = \frac{\sqrt{3}}{4} a^{2}$ $(sqrt(3)a)/2*a/2 = sqrt(3)/4a^2$

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What is area of equilateral triangle?

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