Question

Find the percentage increase in the area of a triangle if each side is doubled.

Answer 1

Step 1: Finding semi-perimeter and area of a triangle

We know that the semi-perimeter of a triangle is given as,

$S=\frac{a+b+c}{2}$, where $a,b,c$ are the sides of the triangle.

Also, area of a triangle, $A=\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

Step 2: Calculate the new semi-perimeter

According to the question, when each side is doubled, the new sides are $2a,2b,2c$.

Hence, the new perimeter is given as,

$S\text{'}=\frac{2a+2b+2c}{2}$

$=2\left(\frac{a+b+c}{2}\right)$

$=2S$

Step 3: Calculate the new area

Hence, the new area is given as,

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

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

$=4\sqrt{s\left(s-a\right)\left(s-b\right)\left(s-c\right)}$

$=4A$

Step 4: Calculate the $\%$ Increase

$\therefore \%$ Change in area$=\frac{A\text{'}-A}{A}\times 100$

$=\frac{4A-A}{A}\times 100$

$=3\times 100=300\%$

Hence, the percentage increase in area is $300\%$.