Spherical Cap

A spherical cap is the region of a Sphere which lies above (or below) a given Plane. If the Plane passes through the Center of the Sphere, the cap is a Hemisphere. Let the Sphere have Radius $R$, then the Volume of a spherical cap of height $h$ and base Radius $a$ is given by the equation of a Spherical Segment (which is a spherical cut by a second Plane)

$$V_{\rm spherical\ segment}={\textstyle{1\over 6}}\pi h(3a^2+3b^2+h^2)$$ (1)

with $b=0$, giving

$$V_{\rm cap}={\textstyle{1\over 6}}\pi h(3a^2+h^2).$$ (2)

Using the Pythagorean Theorem gives

$$(R-h)^2+a^2=R^2,$$ (3)

which can be solved for $a^2$ as

$$a^2=2Rh-h^2,$$ (4)

and plugging this in gives the equivalent formula

$$V_{\rm cap}={\textstyle{1\over 3}}\pi h^2(3R-h).$$ (5)

In terms of the so-called Contact Angle (the angle between the normal to the sphere at the bottom of the cap and the base plane)

$$R-h=R\sin\theta$$ (6)

$$\alpha\equiv \sin^{-1}\left({R-h\over R}\right),$$ (7)

so

$$V_{\rm cap}= {\textstyle{1\over 3}}\pi R^3(2-3\sin\alpha+\sin^3\alpha).$$ (8)

Consider a cylindrical box enclosing the cap so that the top of the box is tangent to the top of the Sphere. Then the enclosing box has Volume

$$V_{\rm box} = \pi a^2h=\pi(R\cos\alpha)[R(1-\sin\alpha)]$$

$$= \pi R^3(1-\sin\alpha-\sin^2\alpha+\sin^3\alpha),$$ (9)

so the hollow volume between the cap and box is given by

$$V_{\rm box}-V_{\rm cap} = {\textstyle{1\over 3}}\pi R^3(1-3\sin^2\alpha+2\sin^3\alpha).$$ (10)

If a second Plane cuts the cap, the resulting Spherical Frustum is called a Spherical Segment. The Surface Area of the spherical cap is given by the same equation as for a general Zone:

$$S_{\rm cap}=2\pi Rh.$$ (11)

See also Contact Angle, Dome, Frustum, Hemisphere, Solid of Revolution, Sphere, Spherical Segment, Torispherical Dome, Zone