Question

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

Answer 1

Answer:

A=s(s−a)(s−b)(s−c)−−−−−−−−−−−−−−−−−√

For equilateral triangle,

a=b=c

s=a+b+c2=3a2

A=3a2(3a2−a)(3a2−a)(3a2−a)−−−−−−−−−−−−−−−−−−−−−−√

A=3a2(a2)(a2)(a2)−−−−−−−−−−√

A=3a416−−−√

A=3√a24

Step-by-step explanation:

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Answer 2

Area=

$\frac{1}{2} bh = \frac{ {a}^{2} \sqrt{3} }{4}$

Explanation:

Equilateral triangles have sides of all equal length and angles of 60°. To find the area, we can first find the height. To find the height, we can draw an altitude to one of the sides in order to split the triangle into two equal 30-60-90 triangles.

Now, the side of the original equilateral triangle (lets call it "a") is the hypotenuse of the 30-60-90 triangle. Because the 30-60-90 triange is a special triangle, we know that the sides are x, x √3, and 2x, respectively.

Thus, a = 2x and x = a/2.
Height of the equilateral triangle = (a√3)/2

Given the height, we can now find the area of the triangle using the equation as shown above.

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