Question

A traffic signal board, indicating ‘SCHOOL AHEAD’, is an equilateral triangle with side a. Find the area of the signal board, using Heron’s formula.If its perimeter is 180 cm, what will be the area of the signal board?​

Answer 1

`900√(3)\:\mathrm{cm^2}\text{ or }\approx 1,558.85\:\mathrm{cm^2}`

Explanation:

Heron's formula can be used to find the area of any triangle and is given by:

`A=√(s(s-a)(s-b)(s-c))`, where
`a`,
`b`, and
`c` are three sides of the triangle and
`s` is the semi-perimeter (
`s=(a+b+c)/(2)`).

By definition, all three sides and angles of an equilateral triangle are equal. Therefore, if the total perimeter is 180 cm, each side must have a length of
`180/ 3=60\text{ cm}`.

The semi-perimeter is therefore:

`s=(60+60+60)/(2)=90\text{ cm}`

Substitute values into Heron's formula to get:

`A=√(90(90-60)(90-60)(90-60)),\\A=√(90\cdot 30\cdot30\cdot 30),\\A=√(2,430,000),\\A=\boxed{900√(3) \:\mathrm{cm^2}}\approx \boxed{1,558.85\:\mathrm{cm^2}}`