# A triangle has sides with lengths: 4, 5, and 7. How do you find the area of the triangle using Heron's formula?

Feb 17, 2016

 4sqrt6 ≈ 9.8 " square units "

#### Explanation:

This is a 2 step process.

step 1 : Calculate half of the perimeter ( s ) of the triangle.
step 2 : Calculate the area (A)

let a = 4 , b = 5 and c = 7

step 1 : s = $\frac{a + b + c}{2} = \frac{4 + 5 + 7}{2} = \frac{16}{2} = 8$

step 2 : $A = \sqrt{s \left(s - a\right) \left(s - b\right) \left(s - c\right)}$

$= \sqrt{8 \left(8 - 4\right) \left(8 - 5\right) \left(8 - 7\right)} = \sqrt{8 \times 4 \times 3 \times 1} = \sqrt{96} = 4 \sqrt{6}$

Feb 17, 2016

$A r e a = 4 \sqrt{6.} u n i t s$

#### Explanation:

$A = A r e a$

$a - b - c = s i \mathrm{de} s$

$s = \frac{a + b + c}{2}$

Heron's formula for the area of the triangle:

color(blue)(A=sqrt(s(s-a)(s-b)(s-c))

In this case color(green)(a=4,b=5,c=7,s=(4+5+7)/2=16/2=8

$\rightarrow A = \sqrt{8 \left(8 - 4\right) \left(8 - 5\right) \left(8 - 7\right)}$

$\rightarrow A = \sqrt{8 \left(4\right) \left(3\right) \left(1\right)}$

$\rightarrow A = \sqrt{8 \left(12\right)}$

rArrcolor(orange)(A=sqrt96=sqrt(16*6)=4sqrt6