# Area of an elliptical arch Calculator

## Calculates the area, length of chord and arch of elliptical arch given two semiaxes and two angles.

angle θ0
angle θ1
 same unit as θ0
semiaxis a
semiaxis b

area of arch S
length of chord c
length of arch L
 $\normal Elliptical\ Arch\\(1)\ area:\\\hspace{20} S=F(\theta_1)-F(\theta_0)-{\large\frac{r_0 r_1}{2}}sin(\theta_1-\theta_0)\\\hspace{20} F(\theta)= {\large\frac{ab}{2}}\left[\theta-tan^{\small-1}\left({\large\frac{(b-a)sin2\theta}{b+a+(b-a)cos2\theta}}\right)\right]\\\hspace{20} r(\theta)^2={\large\frac{a^2b^2}{b^2cos^2\theta+a^2sin^2\theta}}\\\vspace{5}(2)\ elliptical\ arch :\\\hspace{20} L=aE({\large\frac{x(\theta_0)}{a}},k)-aE({\large\frac{x(\theta_1)}{a}},k)\\\hspace{20} x(\theta)=r(\theta)cos\theta,\ k=\sqrt{1-({\large\frac{b}{a}})^2}, \hspace{20} a\ge b,\hspace{10}\frac{\pi}{2}\ge \theta\ge 0\\\hspace{20} E(x,k):\ 2nd\ incomplete\ elliptic\ integral\\\vspace{5}(3)\ elliptical\ chord :\\\hspace{20} c=\sqrt{r(\theta_0)^2+r(\theta_1)^2-2r(\theta_0)r(\theta_1)cos(\theta_1-\theta_0)}\\$

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