Chapter 11.8, Problem 33HP

To determine

To find the relationship between formulae of circumference and area of a circle.

Answer to Problem 33HP

  $a=\frac{c^2}{4\times \pi }$ Where ‘a’ is the area and ‘c’ is the circumference.

Explanation of Solution

Given information:Circumference $c=2\times \pi \times r$ and area $a=\pi \times r^2$ are the formulae for circles.

The common variable of both the formulae of ‘a’ and ‘c’ is the radius ‘r’. On replacing this value of the radius in each of the equations, the relationship between area and circumference can be determined.

Circumference $c=2\times \pi \times r$

or, $r=\frac{c}{2\times \pi }$

Also, area of circle $a=\pi \times r^2$

or, $a=\pi \times r\times r$

Replacing the value of ‘r’ that is derived from the circumference equation and putting it in the equation of area.

  $a=\pi \times \left(\frac{c}{2\times \pi }\right)\times \left(\frac{c}{2\times \pi }\right)$

  $a=\frac{c^2}{4\times \pi }$