Chapter 11, Problem 29RP

To determine

To check: that which has the greater area, a circle with circumference 100 or a square with perimeter 100.

Answer to Problem 29RP

Circle with circumference 100 has the greater area than the square with perimeter 100.

Explanation of Solution

Given Information:

Circumference of a circle $\left(C\right)=100$

Perimeter of a square $\left(P\right)=100$

Formula used:

Circumference of a circle $\left(C\right)=2\pi r$

Area of a circle $\left(A\right)=\pi r^2$

Here, $r$ is the radius of a circle

Perimeter of a square $\left(P\right)=4a$

Area of a square $\left(A_1\right)=a^2$

Here, $a$ is the side of a square

Calculation:

Let $r$ be the radius of a circle

Let $a$ be the side of a square

We know that, Circumference of a circle $\left(C\right)=2\pi r$

  $\begin{array}{l}\therefore 100=2\times \frac{22}{7}\times r\\ \therefore r=\frac{175}{11}\end{array}$

And, area of a circle $\left(A\right)=\pi r^2$

  $\begin{array}{l}\therefore A=\frac{22}{7}\times \frac{175}{11}\times \frac{175}{11}\\ \therefore A=795\end{array}$

Also we know that, Perimeter of a square $\left(P\right)=4a$

  $\begin{array}{l}\therefore 100=4a\\ \therefore a=25\end{array}$

And, area of a square $\left(A_1\right)=a^2$

  $\begin{array}{l}\therefore A_1={25}^2\\ \therefore A_1=625\end{array}$

Hence, the circle has a greater area than the square.