Find the maximum possible area of a triangle whose sides sum to 30 cm. Use Heron's formula in your solution.

Step-by-step solution

1. Let the sides be $a,b,c$ with $a+b+c=30$.
2. The semiperimeter is $s=\frac{30}{2}=15$.
3. Heron's formula gives $A^2=s(s-a)(s-b)(s-c)=15(15-a)(15-b)(15-c)$.
4. For fixed $s$, the product $(15-a)(15-b)(15-c)$ is maximised when the three factors are equal, i.e., $15-a=15-b=15-c$.
5. Thus $a=b=c=10$ cm.
6. Compute the area: $A^2=15(5)(5)(5)=1875$.
7. Therefore $A=\sqrt{1875}=\sqrt{625cdot3}=25\sqrt{3}$.
8. The maximum area is $25\sqrt{3}{\text{cm}}^2$.

Concept tested: Maximum area for fixed perimeter using Heron's formula

Published 13 October 2026 · How this solution was prepared and checked