0
$\begingroup$

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?

$\endgroup$
  • $\begingroup$ Please add some detail, not all terms are common knowledge. $\endgroup$ – vonbrand Sep 17 '15 at 12:23
0
$\begingroup$

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 $s$.

$\endgroup$
0
$\begingroup$

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: $h=R\left( 1-\cos\dfrac{\theta}{2}\right)$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.