Question

Find the area of a triangle whose sides are $a=5,b=8,c=9.$

Short answer

Required area of the triangle is$19.89squareunits.$

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:

$$\begin{gathered} Area=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \\ where, \\ s=\raisebox{1ex}{$a+b+c$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \end{gathered}$$

Step2. Calculation

$$\begin{gathered} Given, \\ a=5,b=8,c=9 \\ then, \\ s=\raisebox{1ex}{$a+b+c$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ =\raisebox{1ex}{$5+8+9$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ =\raisebox{1ex}{$22$}\!\left/ \!\raisebox{-1ex}{$2$}\right. \\ =11 \\ now,Heron\text{'}sformulagivesareaas \\ Area=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)} \\ =\sqrt{11\left(11-5\right)\left(11-8\right)\left(11-9\right)} \\ =\sqrt{11\times 6\times 3\times 2} \\ =\sqrt{396} \\ =19.899\approx 19.89squareunits. \end{gathered}$$