PlanetMath: parts of a ball

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parts of a ball

(Definition)

Let us consider in $\mathbb{R}^3$ a ball of radius and the sphere bounding the ball.

Two parallel planes intersecting the ball separate between them from the ball a spherical segment, which can also be called a spherical frustum (see the frustum). The curved surface of the spherical segment is the spherical zone.
In the special case that one of the planes is a tangent plane of the sphere, the spherical segment is a spherical cap and the spherical zone is a spherical calotte.
The lateral surface of a circular cone with its apex in the centre of the ball divides the ball into two spherical sectors.

The distance of the two planes intersecting the ball be is called the height.

The volume of the spherical cap is obtained from

$\displaystyle V \,=\, \pi h^2\left(r\!-\!\frac{h}{3}\right)$

and the area of the corresponding spherical calotte and also a spherical zone from

$\displaystyle A \,=\, 2\pi rh.$

The volume of a spherical segment can be got as the difference of the volumes of two spherical caps.

The volume of a spherical sector may be calculated from

$\displaystyle V \,=\, \frac{2}{3}\pi r^2h,$

where

is the height of the spherical cap of the spherical sector.

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"parts of a ball" is owned by pahio. [ full author list (2) ]

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Other names:

parts of ball, parts of sphere

Also defines:

spherical segment, spherical frustum, spherical cap, spherical calotte, spherical sector

This object's parent.

Attachments:: area of spherical zone (Derivation) by pahio
volume of spherical cap and spherical sector (Theorem) by pahio

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Cross-references: difference, area, volume, distance, divides, apex, circular cone, lateral surface, tangent plane, surface, frustum, planes, parallel, sphere, radius, ball
There are 4 references to this entry.

This is version 7 of parts of a ball, born on 2008-08-10, modified 2008-08-19.
Object id is 10932, canonical name is PartsOfABall.
Accessed 766 times total.

Classification:

51M05 (Geometry :: Real and complex geometry :: Euclidean geometries and generalizations)

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