Spherical Cap

Finds the volume and surface area of the piece left when a sphere is cut by a plane. It gives the liquid held in a spherical tank and the size of a domed roof. The radius of the circular cut is also shown.

One straight cut through a sphere leaves a larger piece and a smaller one. The piece removed is a spherical cap. It is the shape behind the liquid held in a spherical tank and the area of a domed roof.

V = \dfrac{\pi h^2 (3r - h)}{3} \qquad S = 2\pi r h

$V$ is the volume of the cap
$S$ is its curved surface area, not counting the flat circular cut
$r$ is the radius of the sphere
$h$ is the height of the cap

The radius of the circular cut is $\sqrt{h(2r - h)}$ , which follows from the Pythagorean theorem once you notice the centre of the sphere lies $r - h$ from the cutting plane.

The curved area depends only on the height

Look closely at $2\pi r h$ . It contains the height of the cap and nothing about where on the sphere the cut was made.

A band sliced near the pole and a band of the same thickness sliced at the equator therefore have identical curved areas. Near the pole the ring is small but the surface is steeply tilted; near the equator the ring is large but the surface is nearly upright. The two effects cancel exactly. Archimedes discovered this, and the sphere's total area of $4\pi r^2$ follows from it directly.

Example

The default input cuts a cap of height 2 from a sphere of radius 5.

The volume is $\pi \times 4 \times (15 - 2) \div 3$ , about 54.4543. The curved area is $2\pi \times 5 \times 2$ , about 62.8319. The radius of the cut is $\sqrt{2 \times 8}$ , exactly 4, and the total surface including the flat face is about 113.0973.

Setting the height to the full diameter of 10 gives a volume of about 523.5988 and a curved area of about 314.1593 — precisely the sphere of radius 5. Setting it to 5, equal to the radius, gives a hemisphere, with exactly half the sphere's volume and half its curved area.

Points to watch

The height cannot exceed the diameter; beyond that there is no cap to speak of.

When gauging a spherical tank, if the liquid rises above the centre it is easier to treat the empty space above it as the cap and subtract from the whole sphere. Compute the liquid directly as a cap only while it sits in the lower half.

The curved area excludes the circular cut. Use the curved figure for roofing material on a dome, and the total for painting a solid object that includes its flat face.