# Spherical Segment

Spherical segment is a solid bounded by two parallel planes through a sphere. In terms of spherical zone, spherical segment is a solid bounded by a zone and the planes of a zone's bases.

**Properties of Spherical Segment**

- The
*bases*of a spherical segment are the sections made by the parallel planes. The radii of the lower and upper sections are denoted by a and b, respectively. If either a or b is zero, the segment is of one base. If both a and b are zero, the solid is the whole sphere. - If one of the parallel planes is tangent to the sphere, the solid thus formed is a
*spherical segment of one base*. - The spherical segment of one base is also called
*spherical cap*and the two bases is also called*spherical frustum*. - The
*altitude*of the spherical segment is the perpendicular distance between the bases. It is denoted by h.

**Formulas for Spherical Segment**

**Area of lower base, A**

_{1}

**Area of upper base, A**

_{2}

**Area of the zone, A**

_{zone}

**Total Area, A**

The total area of

*segment of a sphere*is equal to area of the zone plus the sum of the areas of the bases.

$A = A_{zone} + A_1 + A_2$

$A = 2\pi Rh + \pi a^2 + \pi b^2$

**Volume, V**

The volume of

*spherical segment of two bases*is given by

*spherical segment of one base*is given by

The formula for the volume of one base can be derived from volume of two bases with b = 0. Consider the following diagram:

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

$a^2 + (R^2 - 2Rh + h^2) = R^2$

$a^2 = 2Rh - h^2$

Substitute a^{2} = 2Rh - h^{2} and b = 0 to the formula of spherical segment of two bases

$V = \frac{1}{6}\pi h(3a^2 + 3b^2 + h^2)$

$V = \frac{1}{6}\pi h \, [ \, 3(2Rh - h^2) + 3(0^2) + h^2 \, ]$

$V = \frac{1}{6}\pi h \, [ \, 6Rh - 3h^2 + h^2 \, ]$

$V = \frac{1}{6}\pi h \, [ \, 6Rh - 2h^2 \, ]$

$V = \frac{1}{6}\pi h (2h) [ \, 3R - h \, ]$

$V = \frac{1}{3}\pi h^2 (3R - h)$ (*okay!*)

Note also that the volume of *segment of a sphere* of altitude h and radii a and b is equal to the volume of a sphere of radius h/2 plus the sum of the volumes of two cylinders whose altitudes are h/2 and whose radii are a and b, respectively.