Question Number 187452 by Rupesh123 last updated on 17/Feb/23
![](https://www.tinkutara.com/question/29374.png)
Answered by mr W last updated on 17/Feb/23
![side length of octagon = a area of octagon A_8 =(a+2×(a/( (√2))))^2 −a^2 =2(1+(√2))a^2 black area A_(black) =a^2 fraction of black area=(a^2 /(2(1+(√2))a^2 ))≈20.71%](https://www.tinkutara.com/question/Q187455.png)
$${side}\:{length}\:{of}\:{octagon}\:=\:{a} \\ $$$${area}\:{of}\:{octagon}\:{A}_{\mathrm{8}} =\left({a}+\mathrm{2}×\frac{{a}}{\:\sqrt{\mathrm{2}}}\right)^{\mathrm{2}} −{a}^{\mathrm{2}} =\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right){a}^{\mathrm{2}} \\ $$$${black}\:{area}\:{A}_{{black}} ={a}^{\mathrm{2}} \\ $$$${fraction}\:{of}\:{black}\:{area}=\frac{{a}^{\mathrm{2}} }{\mathrm{2}\left(\mathrm{1}+\sqrt{\mathrm{2}}\right){a}^{\mathrm{2}} }\approx\mathrm{20}.\mathrm{71\%} \\ $$
Commented by Rupesh123 last updated on 17/Feb/23
Good
Answered by HeferH last updated on 17/Feb/23
![Total area = ((P∙a)/2) a: apothem P: perimeter let x be the side of the octagon Total area = 8x ∙ (((x + x(√2))/2)) ∙(1/2) area shaded = x^2 ((Total area)/(area shaded)) = (x^2 /((8x^2 (1 +(√2)))/4)) = (x^2 /(2x^2 (1+(√2))))= (1/(2+2(√2)))](https://www.tinkutara.com/question/Q187467.png)
$${Total}\:{area}\:=\:\frac{{P}\centerdot{a}}{\mathrm{2}} \\ $$$$\:{a}:\:{apothem} \\ $$$$\:{P}:\:{perimeter} \\ $$$$\:{let}\:{x}\:{be}\:{the}\:{side}\:{of}\:{the}\:{octagon} \\ $$$$\:{Total}\:{area}\:=\:\mathrm{8}{x}\:\centerdot\:\left(\frac{{x}\:+\:{x}\sqrt{\mathrm{2}}}{\mathrm{2}}\right)\:\centerdot\frac{\mathrm{1}}{\mathrm{2}} \\ $$$$\:{area}\:{shaded}\:=\:{x}^{\mathrm{2}} \\ $$$$\frac{{Total}\:{area}}{{area}\:{shaded}}\:=\:\:\frac{{x}^{\mathrm{2}} }{\frac{\mathrm{8}{x}^{\mathrm{2}} \left(\mathrm{1}\:+\sqrt{\mathrm{2}}\right)}{\mathrm{4}}}\:=\:\frac{{x}^{\mathrm{2}} }{\mathrm{2}{x}^{\mathrm{2}} \left(\mathrm{1}+\sqrt{\mathrm{2}}\right)}=\:\frac{\mathrm{1}}{\mathrm{2}+\mathrm{2}\sqrt{\mathrm{2}}}\: \\ $$
Commented by Rupesh123 last updated on 17/Feb/23
Excellent!