# Volume and Surface Area of Spherical Cap Calculator

## Spherical Cap

A spherical cap is defined as as a portion of the sphere cut by a plane.## Formulas for the Volume and Surface Area of a Spherical Cap

The volume of a spherical cap of height \( h \) is given by the formula\[ \displaystyle \text{Volume} = \dfrac{\pi}{3}( 3 Rh^2-h^3) \]

The area of the spherical cap is given by

\[ \displaystyle \text{Area} = 2\pi R h \]

The radius \( r \) of the circle whose diameter in \(AB\) and the angle \( \alpha \) shown in the figure above are given by

\[ \displaystyle \text{r} = \sqrt {R^2 - (R-h)^2 } \]

\[ \displaystyle \alpha = \arcsin \left(\dfrac{r}{R}\right) \]

## How to use the calculator

1 - Enter the radius \( R \) of the sphere from which the spherical cap is cut and the height \( h \) as positive real numbers, with \( R \ge h \) and press "calculate". The outputs are volume \( V_{cap} \) and the lateral area \(A_{cap} \) of the spherical cap, the radius of the cap \( r \), angle \( \alpha \), the ratio \( \dfrac{V_{cap}}{V_{sphere}} \) of the volume of the cap to that of the sphere and the ratio \( \dfrac{A_{cap}}{A_{sphere}} \) of the area of the cap to that of the area of the shpere .## Outputs

Enter the radius \( R \) of the sphere from which the spherical cap is cut and the volume \( V \) of the spherical cap as positive real numbers, with \( V \) is less than half the volume of the sphee of radius \( R \) and press "calculate". The outputs are the height \( h \) and the lateral area \(A_{cap} \) of the spherical cap, the radius of the cap \( r \), angle \( \alpha \), the ratio \( \dfrac{V_{cap}}{V_{sphere}} \) of the volume of the cap to that of the sphere and the ratio \( \dfrac{A_{cap}}{A_{sphere}} \) of the area of the cap to that of the area of the shpere .

## Outputs

## More References and links

Volume of a Spherical Cap .Sectors and Circles Problems .

Circles, Sectors and Trigonometry Problems with Solutions and Answers .

Online Geometry Calculators and Solvers .