# Given a square with side s calculate the area of the curved square, (purple shade area in the figure to the left)? The 2nd figure is to the right is sketched to help guide your thinking?

Nov 6, 2016

$0.3151 {s}^{2}$

#### Explanation:

$A B = B T = A T = s , \implies \Delta A B T$ is equilateral.
$\implies \angle A B T = {60}^{\circ}$
$\angle A B V = {45}^{\circ} , \implies \angle U B V = {15}^{\circ} , \implies \angle U B T = {30}^{\circ}$
$\implies U T = 2 \cdot s \cdot \sin {15}^{\circ} = 0.5176 s$
Let the shaded area (U->T->U) be ${A}_{S}$
${A}_{S} = \pi {s}^{2} \left(\frac{30}{360}\right) - \left(\frac{1}{2} {s}^{2} \cdot \sin 30\right) = 0.0118 {s}^{2}$

$U W X T$ is a square with sides $= 0.5176 s$
Let area of $U W X T$ be ${A}_{Q}$
${A}_{Q} = {\left(0.5176 s\right)}^{2} = 0.2679 {s}^{2}$

Let the area of the curved square in your diagram be ${A}_{C}$
${A}_{C} = {A}_{Q} + 4 \cdot {A}_{S}$
${A}_{C} = 0.2679 {s}^{2} + 4 \times 0.0118 {s}^{2} = 0.3151 {s}^{2}$