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How do you derive the formula for the area of an equilateral triangle?

1 Answer

Jan 13, 2016

Area of an equilateral triangle with sides of length $s$ $s$
$XXX = \frac{\sqrt{3}}{4} \cdot s^{2}$ $color(white)("XXX")=sqrt(3)/4*s^2$
(Derivation below)

Explanation:

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The common formula for the area of a triangle is
$XXX {Area}_{△} = \frac{base \times height}{2}$ $color(white)("XXX")"Area"_triangle = ("base" xx "height")/2$

For an equilateral triangle with sides of length $s$ $s$

$base = s$ $"base" = s$

and the $height$ $"height"$ can be calculated using the Pythagorean Theorem as:

$h e i g h t = \sqrt{s^{2} - {(\frac{s}{2})}^{2}}$ $height" = sqrt(s^2-(s/2)^2)$

$XXXXX = \sqrt{\frac{3 s^{2}}{4}}$ $color(white)("XXXXX")=sqrt((3s^2)/4)$

$XXXXX = \frac{s}{2} \sqrt{3}$ $color(white)("XXXXX")=s/2sqrt(3)$

and therefore the area is
$XXX {Area}_{△} = s \times \frac{\frac{s}{2} \sqrt{3}}{2}$ $color(white)("XXX")"Area"_triangle = s xx (s/2sqrt(3))/2$

$XXXXXXX = \frac{\sqrt{3}}{4} s^{2}$ $color(white)("XXXXXXX")=sqrt(3)/4 s^2$

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