find the area of an equilateral triangle with a perimeter of 45?

Question

the answer is (bear with me now) 225*sqrt of 3 over (divided by) 4... i just need to know how to get there

Answer 1

If the perimeter is 45, each side is 15.

The area of a triangle is 1/2 base * height.

The base of this triangle is 15.

The height can be measured as a line that bisects the top angle of the triangle, making two smaller right triangles where the hypotenuse is 15 and the base is 7.5 (half of the side).

Using the pythagorean theorem:

15^2 = 7.5^2 + x^2

225 * 3/4 = x^2
15/2 * sqrt(3) = x

The area is that height, times the base (15), divided by 2,

15 * 15/2 sqrt(3) / 2 = 225/4 sqrt(3)

Answer 2

Since you know the three sides (it is equilateral, therefore each side is 45/3 =15) and you know the perimeter (45) the choice is to use Heron's formula for area in terms of the semiperimeter of the triangle (half the perimeter) and each side.

For a well formed triangle of sides a, b, c and semiperimeter s = (1/2)(a + b + c) the area S of the triangle is given by the formula

S = sqrt[ s (s - a)(s - b) (s -c )]

Plugging in

S = sqrt[(45/2)(45/2 -15)^3]
= sqrt[ (45/2) (15/2)^3 ]
= (15/2) sqrt[ (45/2) (15/2) ] = (15/2) sqrt( 3 (15^2)/(2^2) ]
= (15/2)(15/2)sqrt (3)
= (15^2/2^2) sqrt(3)
= (225/4) sqrt(3)

Answer 3

If the perimeter is 45, and it is equilateral, that means each side is 15. The area of a triangle is 1/2BH
So...

1/2(15)*15= 112.5