# What is the hatched area of ​​an ellipse?

ellipse described about the circle in which a regular pentagon is constructed mapped on an ellipse

The surface can be calculated from my formula

$A=\frac{a.b.\pi.\alpha}{360}$

Total area will be an ellipse

Area n work will be

$An=\frac{b.\sin\alpha.a \cos\alpha}{2}+\frac{b.\sin\alpha.a(1-\cos\alpha)}{2}$

$An=\frac{a.b}{2}(\sin\alpha\cos\alpha+\sin\alpha-\sin\alpha\cos\alpha)$

$An=\frac{a.b}{2}\sin\alpha=\frac{a.b}{2}\sin(\frac{360}{n})$

How is this n part of it multiplied by n

$A=a.b.\pi$

$\frac{n}{2}\sin(\frac{360}{n})$ look

It is $\frac{1}{5}$ of the area of the ellipse, which is $\frac{\pi}{5}ab$. The mapping from the circle to the ellipse by scaling along the $y$-axis maps the points from the pentagon, to the points on the ellipse. This mapping preserves relative areas, and since the pentagon divides the circle into $5$ equal sectors, the mapped points divide the ellipse into $5$ equal sectors.
The area of an ellipse sector from $0$ to $\theta$ degrees is $a b \theta/2$. The angle $\theta$ is measured along the circle with radius $a$. If a regular n-gon in inscribed into this circle then the angle of one sector is $\theta = 2\pi/n$. Hence, the area of an ellipse sector corresponding to one n-gon sector is $\pi a b / n$, for a pentagon $\pi a b / 5$.