The perimeter of a semi-circle is same as the perimeter of a square . Than whether the area of the semi-circle is greater than area of square or vice versa????
Hello, saroja!
A semicircle of radius $\displaystyle r$ has a perimeter of $\displaystyle \pi r + 2r$The perimeter of a semicircle is same as the perimeter of a square.
Then: .[area of semi-circle] .(> or <) [area of square] ?
A square of side $\displaystyle x$ has a perimeter of $\displaystyle 4x$
. . Hence, we have: .$\displaystyle (\pi+ 2)r \;=\;4s\quad\Rightarrow\quad \frac{r}{s} \:=\:\frac{4}{\pi+2}$
Square both sides: .$\displaystyle \frac{r^2}{x^2} \;=\;\frac{16}{(\pi+2)^2} $
Multiply both sides by $\displaystyle \frac{1}{2}\pi\!:\;\;\frac{\frac{1}{2}\pi r^2}{x^2} \;=\;\frac{8\pi}{(\pi+2)^2} $
We have: .$\displaystyle \frac{\text{Area of semicircle}}{\text{Area of square}} \;=\;\frac{8\pi}{(\pi+2)^2} \;\approx\;0.95$
Therefore: .$\displaystyle \text{[Area of semicircle]} \;\;\bf{{\color{red}<}} \;\; \text{[Area of square]} $