How do I express the area A of an equilateral triangle as a function of the length S of its sides?

Answer 1

Take an equilateral triangle and bisect it. You will create two right triangles with hypotenuse S, and base S/2. By the Pythagorian Theorem, the height is √(S^2 - (S/2)^2 = √([3/4]s^2) = (√3)S/2

So (√3)S/2 is the height. (S·(√3)S/2)/2 = √3/4*S^2

Answer 2

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For side x of an equilateral triangle, A = x^2 sqrt(3) / 4 Split the triangle into two congruent triangles with an altitude from one vertex, which will be the height of the equilateral triangle. This will create two 30-60-90 triangles with base x/2, hypotenuse x, and altitude x/2 * sqrt(3) from the properties of a 30-60-90 triangle (sides in ratio of 1-2-sqrt(3)) Now for the original triangle, we have base x and height x/2 * sqrt(3) A = 1/2 (b) (h) = 1/2 (x) (x/2 * sqrt(3)) = x^2 sqrt(3) / 4

Answer 3

Starting from the fact that the all angles in an equilateral triangle (in euclidean geometry) measure 60º, and the fact that A=1/2 b*h can be transformed by simple trigonometry into A=1/2b*c(sin(x)) where c is any other side of the triangle and x is the angle between c and b. In this particular case you get A=1/2(s)(s*sin(60)) that is A=1/2(s^2)(Sqrt(3)/2). So finally A=1/4(Sqrt(3)*s^2) where s is the lenght of the side.

Answer 4

the formula is A = sqrt(3)/4 * S^2

proof: an equalateral with the side of S

Area =1/2 * base * height

the base is S
Since the interior angle measure of the equalateral triangle are all 60 degrees, when you draw an altitude, that altitude will bisect the base forming two 30-60-90 triangles with the base is S/2.

the long leg (which is also the altitude) is sqrt(3) times the short leg (which is S/2)

so A = 1/2 * S * sqrt(3) (S/2)

A = sqrt(3)/4 * S^2

Answer 5

The angle between sides is 60°, so its sine is (√3)/2 ~= 0.866.
Therefore, the base is S and the height is 0.866S, so the area is approximately 0.433S² or S²sin(60°)/2 or S²(√3)/4

Answer 6

H² = S² - (0.5S)²
=> H = √[S² - 0.25S²]
=> H = √ (0.75S²)

area A = 1/2 B X H
area A = 1/2[S X √ (0.75S²)]
area A = 1/2[S X S √0.75]
area A = 1/2[ S²√0.75]
area A = 1/4*√3*S²

Answer 7

Area as a function S

A(s) = (s*h)/2 s denotes the base
find the height in terms of s
a^2+b^2 = c^2
(s/2) +b^2 = s^2

s^2/4 + b^2 = s^2
b^2 = s^2 - (s^2/4)
b = sqrt(s^2 - (s^2/4)) ----> this is the height so
A(s) = s * ( sqrt(s^2 - (s^2/4)) ) divided by 2

Answer 8

as h = 1/2 sqrt(3) *a
A= 1/4a^2*sqrt(3)