Question

A triangle has sides with lengths: 7, 6, and 8. How do you find the area of the triangle using Heron's formula?

Answer 1

Substitute the lengths into the formula and calculate.

Explanation:

Heron's formula states that the area of a triangle whose sides have lengths a, b, and c is
`A={\sqrt {s(s-a)(s-b)(s-c)}}`
where s is the semiperimeter of the triangle; that is,
`s={\frac {a+b+c}{2}}`
In this example `s = (7 + 6 + 8)/2 = 21/2`
`s - a = 21/2 - 7 = (21- 7*2)/2 = 7/2`
`s - b = 21/2 - 6 = (21 - 12)/2 - 9/2`
`s - c = 21/2 - 8 = 5/2`
`A = sqrt(21/2 * 7/2 * 9/2 * 5/2)`
`A = sqrt((21*7*9*5)/16)`
`A= sqrt(3*7*7*3*3*5)/4`
`A=7*3*sqrt(3*5)/4`
`A=21*sqrt(15)/4`