Question

The perimeter of an equilateral triangle is 60m. The area is
A) \[10\sqrt 3 {m^2}\]
B) \[15\sqrt 3 {m^2}\]
C) \[20\sqrt 3 {m^2}\]
D) \[100\sqrt 3 {m^2}\]

Answer 1

Hint: first using the formula for the perimeter of an equilateral triangle we get the length of the side of a triangle then using the length of a side we will get an area of an equilateral triangle. As the perimeter of the triangle is the sum of all its sides while the area of an equilateral triangle is given as \[\dfrac{{\sqrt 3 }}{4}{a^2}\]. Using both the given concepts information calculate the length of sides and then put the length in the formula of area of a triangle.

Complete step by step solution: Given: perimeter of an equilateral triangle is 60m
Perimeter of an equilateral triangle \[ = 3a\]
\[ \Rightarrow 3a\]\[ = 60\]
On dividing the whole equation by 3 we get,
\[ \Rightarrow \]\[a = \dfrac{{60}}{3}\]
On simplification we get,
\[ \Rightarrow \]\[a = 20\]m
Area of an equilateral triangle \[ = \] \[\dfrac{{\sqrt 3 }}{4}{a^2}\]
On substituting value of a we get,
\[ = \]\[\dfrac{{\sqrt 3 }}{4}{\left( {20} \right)^2}\]
On simplification we get,
\[ = \]\[100\sqrt 3 {m^2}\]
Thus, Area of an equilateral triangle \[100\sqrt 3 {m^2}\]

Hence, option D. \[100\sqrt 3 {m^2}\] is correct answer.

Note: Equilateral triangle: - An equilateral triangle is a triangle in which all three sides are equal. It is also known as Equiangular in Euclidean geometry that is all the three angles are also congruent to each other and are each \[60^\circ \]. Remember the formula of equilateral triangle as \[\dfrac{{\sqrt 3 }}{4}{a^2}\].