Calculate area and perimeter

Question

I need 2 ways to calculate the area and perimeter of the shaded region

I've drawn it several times but the angles do not fit me

A) by elementary geometry

B) by integral calculation

Answer 1

By drawing a reasonable amount of auxiliary lines and circles it is pretty clear the area of the shaded region equals the area of an equilateral triangle with side length $2$, hence the wanted area is $\color{red}{\sqrt{3}}$.

With an integral:

$$ A = 2\int_{-2}^{1}\left|\sqrt{3}-\sqrt{4-x^2}\right|\,dx = \color{red}{\sqrt{3}}. $$

Answer 2

Hint. Let $E$ and $F$ be the intersections of the two semi-circles ($E$ on the left and $F$ on the right) and let $M$ be the midpoint of $AD$. Then the triangles $\triangle AEM$, $\triangle EFM$, and $\triangle FDM$ are all equilateral with sides of length $2$.