If the circumference of a circle is increased by 20%, then the area of the circle will be increased by:
Answer & explanation
Correct answer: option 2
The correct answer is Option (2) → 44%
The circumference \(2\pi r\) changes in the same proportion as the radius, but the area \(\pi r^2\) changes with the square of the radius.
A 20% larger circumference means the radius becomes \(1.2r\)
New area with radius \(1.2r\): \(\pi(1.2r)^2=1.44\pi r^2\)
Increase \(=1.44-1=0.44\) of the old area, that is, 44%
Shortcut: For a quantity that depends on the square of a length, a rise of \(x\%\) in the length is like two successive rises of \(x\%\): the total rise is \(x+x+\frac{x\times x}{100}\%\).
With \(x=20\): \(\displaystyle 20+20+\frac{20\times20}{100}=44\%\)