Question Number 222153 by fantastic last updated on 19/Jun/25
Commented by fantastic last updated on 19/Jun/25
$${area}\:{if}\:{side}\:{length}\:{of}\:{square}\:{is}\:{a} \\ $$
Answered by mr W last updated on 19/Jun/25
Commented by mr W last updated on 19/Jun/25
$${A}_{\mathrm{1}} =\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{3}\:\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}+\frac{\pi}{\mathrm{2}}−\mathrm{2}\right){a}^{\mathrm{2}} \\ $$$$\left({see}\:{Q}\mathrm{222072}\right) \\ $$$${A}_{\mathrm{3}} =\mathrm{2}\left(\frac{\pi{r}^{\mathrm{2}} }{\mathrm{4}}−\frac{{r}^{\mathrm{2}} }{\mathrm{2}}\right)=\frac{\mathrm{1}}{\mathrm{4}}\left(\frac{\pi}{\mathrm{2}}−\mathrm{1}\right){a}^{\mathrm{2}} \\ $$$${A}_{{shaded}} ={A}_{\mathrm{1}} −{A}_{\mathrm{3}} \\ $$$$\:\:\:\:\:\:\:\:\:\:\:\:=\frac{\mathrm{1}}{\mathrm{4}}\left(\mathrm{3}\:\mathrm{tan}^{−\mathrm{1}} \frac{\mathrm{1}}{\mathrm{2}}−\mathrm{1}\right){a}^{\mathrm{2}} \approx\mathrm{0}.\mathrm{098}{a}^{\mathrm{2}} \\ $$
Commented by fantastic last updated on 19/Jun/25
$${lovely},{lovely}! \\ $$