Question

Among the following shapes with equal perimeter, which one has the largest area?

• A. A square
• B. An equilateral triangle
• C. A circle
• D. A regular pentagon

Answer 1

The correct option is C A circle

For a given perimeter of length ‘a’ :

i.) Length of side of a square = $\frac{a}{4}$
So, Area of the Square = $\frac{a^2}{16}$ = $0.0625a^2$

ii.) Length of side of an equilateral triangle = $\frac{a}{3}$
Area of the equilateral triangle = $\frac{\sqrt{3}}{4}\times a^2$ = $0.048a^2$

iii.) Length of radius of a circle = $\frac{a}{2\times \pi }$
Area of the circle = $(\frac{a}{2\times \pi }{)}^2$= $0.0796a^2$

iv.) Length of side of a regular pentagon = $\frac{a}{5}$
Area of the regular pentagon $\frac{1}{4}\times \sqrt{5(5+2\sqrt{5})}\times a^2$ = $0.069a^2$