Question

Out of the following shapes having equal perimeters, which is the one having the largest area :

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

Answer 1

The correct option is A circle

$LetthegivenperimeterbeP.Wecalculatetheareasofthegivenfiguresandcompare.OptionA⟶circle.Theperimeter\left(circumference\right)=P⟹2\pi r=P(whenr=radiusofthecircle.)⟹r=\frac{P}{2\pi }\therefore ar.circle=\pi r^2=\pi \times {\left(\frac{P}{2\pi }\right)}^2=\frac{P^2}{4\times 3.14}=\frac{P^2}{12.56}.OptionB⟶Equilateraltriangle.Oneside=\frac{P}{3}.\therefore area=\frac{\sqrt{3}}{4}{side}^2=\frac{\sqrt{3}}{4}\times {\left(\frac{P}{3}\right)}^2=\frac{P^2}{20.77}.OptionC⟶Squareside=\frac{P}{4}\therefore area=\frac{P}{4}\times \frac{P}{4}=\frac{P^2}{16}.OptionD⟶Regularpentagon.side=\frac{P}{5}.\therefore ar.pentagon=\frac{1}{4}\sqrt{5(5+2\sqrt{5})}\times {side}^2=\frac{1}{4}\sqrt{5(5+2\sqrt{5})}\times {\left(\frac{P}{5}\right)}^2=\frac{P^2}{19.95}.\therefore comparingtheareaswegetthear.circleisthegreatest.Ans-OptionA$