Find Sagitta from Segment Area and Radius

Question

Is it possible to calculate the Sagitta, knowing the Segment Area and Radius? Alternatively, is there a way to calculate the Chord Length, knowing the Segment Area and Radius?

Answer 1

Ler $R$ be the radius, $A$ the area of the segment, $s$ the sagitta and $\alpha=\arcsin\dfrac{R-s}{R}$ half the angle subtended by the segment. Then $$ A=2\int_{R-s}^R\sqrt{R^2-x^2}\,dx=2\,R^2\int_\alpha^{\pi/2}\cos^2t\,dt=\frac{R^2}2\bigl(\pi+\sin(2\,\alpha)-2\,\alpha\bigr). $$ This is a transcendental equation in $\alpha$, to be solved by numerical methods. Once you know $\alpha$ you can compute $s$.

Answer 2

If $R$ is the radius of the circle, than the area of a circular sector subtended by a central angle $\theta$ is: $A_S=R^2 \dfrac{\theta}{2}$, and the area of the triangle defined by the two radii and the chord is :$A_T=\dfrac{1}{2}R^2 \sin \theta$ , so the area of the segment is $A=A_S-A_T=\dfrac{R^2}{2}\left( \theta-\sin \theta\right)$. From this you can find $\theta$ (but, generally, only with an approximate solution), than you can find the sagitta $h$ from: $h=R\left( 1-\cos\dfrac{\theta}{2}\right)$

Answer 3

Chord subtends $2\theta $ at circle center. $$ s =R ( 1- \cos \theta);\; A= \theta R^2;\;\to s= R(1- \cos \frac{A}{R^2}). $$