# Given an octagon inscribed in a square of side, s express the shaded area i terms of s? Hint the intersecting circles quarter circles have radius that are related to the diagonal of the square...

## Oct 30, 2016

$\left(\frac{1}{2} \pi - 1\right) {s}^{2}$

#### Explanation: Given $A B = C D = s , \implies C B = \sqrt{2} s$,
$\implies x = \frac{\sqrt{2} s}{2} = \frac{s}{\sqrt{2}}$
Area $\Delta O C D = \frac{1}{4} {s}^{2}$

The two green areas in $\Delta O C D$ are the same.

One green area ${A}_{G} = \frac{1}{4} {s}^{2} - \pi {\left(\frac{s}{\sqrt{2}}\right)}^{2} \cdot \frac{45}{360}$
${A}_{G} = \left(\frac{1}{4} - \frac{1}{16} \pi\right) {s}^{2}$

Let the black area in $\Delta O C D$ be ${A}_{B}$
$\implies {A}_{B} = \frac{1}{4} {s}^{2} - 2 \cdot {A}_{G}$
${A}_{B} = \frac{1}{4} {s}^{2} - 2 \left(\frac{1}{4} - \frac{1}{16} \pi\right) {s}^{2}$
${A}_{B} = \left(\frac{1}{4} - \frac{1}{2} + \frac{1}{8} \pi\right) {s}^{2}$
${A}_{B} = \left(\frac{1}{8} \pi - \frac{1}{4}\right) {s}^{2}$

Now, let the shaded area in your diagram be ${A}_{S}$
$\implies {A}_{S} = 4 \cdot {A}_{B}$
${A}_{S} = 4 \left(\frac{1}{8} \pi - \frac{1}{4}\right) {s}^{2}$
${A}_{S} = \left(\frac{1}{2} \pi - 1\right) {s}^{2} = 0.5708 {s}^{2}$