Question

Find the area of a triangle whose sides are$a=12,b=13,c=5$.

Short answer

Required area of the triangle is 30 square units.

Step-by-step solution

Step 1. Given information

The area of a triangle with sides a, b, and c is determined by the Heron's formula which is given as:

$Area=\sqrt{s(s-a)\left(s-b\right)\left(s-c\right)}$

where, $s=\raisebox{1ex}{$a+b+c$}\!\left/ \!\raisebox{-1ex}{$2$}\right.$

Step 2. Calculation

Given,

$$\begin{gathered} a=12,b=13,c=5 \\ Then, \\ s=\raisebox{1ex}{$a+b+c$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ =\raisebox{1ex}{$12+13+5$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ =15 \\ Now,Heron\text{'}sFormulagivesareaas \\ Area=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \\ =\sqrt{15\left(15-12\right)\left(15-13\right)\left(15-5\right)} \\ =\sqrt{15\times 3\times 2\times 10} \\ =\sqrt{900} \\ =30squareunits \end{gathered}$$