Question

What is the area of an equiangular triangle with perimeter 36?

Answer 1

Hint: Here in this question we have been asked to find the area of an equiangular triangle with the perimeter given as 36. For answering this question we will first find the side length of the given triangle and then use the formula for finding the area of an equilateral triangle given as $\dfrac{\sqrt{3}}{4}{{a}^{2}}$ .

Complete step-by-step solution:
Now considering the question we have been asked to find the area of an equiangular triangle with the perimeter given as 36.
From the basic concepts we know that the formula for finding area of an equilateral triangle will be given as $\dfrac{\sqrt{3}}{4}{{a}^{2}}$ where $a$ is the length of a side of the triangle.
Now we can say that the given triangle is equiangular that means all angles are equal and that implies that all sides of the triangle are equal so we can say that the triangle is equilateral.
Now as we have the perimeter of the triangle from the question we need to use that and find the side length we know that the perimeter of any shape is the length of the boundary of the shape that means the perimeter of an equilateral triangle is three times the length of the side.
Therefore the length of the side will be equal to $\dfrac{36}{3}=12$ .
Now we can say that the area of the triangle will be given as $ \dfrac{\sqrt{3}}{4}{{\left( 12 \right)}^{2}}=36\sqrt{3}$ .
Therefore we can conclude that the area of an equiangular triangle with the perimeter given as 36 will be given as $36\sqrt{3}$.

Note: While answering this type of questions we should be sure with the calculations that we are going to perform in between the steps. Someone can make a mistake during the calculation and consider it as $ \dfrac{\sqrt{3}}{4}{{\left( 12 \right)}^{2}}=39\sqrt{3}$ which will lead them to end up having a wrong conclusion.