Spherical Segment Formula

If we cut a basketball with a pair of two knives parallely, then the solid that is defined by the cutting is the spherical segment. You can say that the spherical cap has been truncated, and so it can be called as a spherical frustum.Those spherical segment is called spherical zone, where we exclude the bases.

spherical segment

The formula that we use here are volume and surface area:

\[\large A=2\pi Rh\]

\[\large V=\frac{\pi h}{6}(3r_{1}^{2}+r_{2}^{2}+h^{2})\]

Solved Example

Question: What will be the volume of a segment of a sphere,the radius of the base being 9.2 cms, the radius of sphere 11 cms and height is 7 cms ?

Solution:

Using the formula: $V=\frac{\pi h}{6}(3r_{1}^{2}+r_{2}^{2}+h^{2})$

$V=\frac{\pi \times 7}{6}(3\times 9.2^{2}+3\times 11^{2}+7^{2})$

$= 3.66 \times 665.92$

$=2437.26\;cm^{2}$


Practise This Question

Pinku was a hard working student who used to learn without understanding. He was asked to construct a triangle say ABC and was given the base length of the triangle BC, one of the base angles say  B  and the sum of the other two sides (AB + AC). He went about constructing the triangle in the following way:

He drew the base BC with the given dimension, drew the  B along the ray BX with the angle known to him already. He then took B as centre and (AB + AC) as radius and cuts an arc on the ray BX intersecting the ray at D.

He then joins D to C. He then draws a perpendicular bisector of the line DC and the perpendicular bisector intersecting on the ray intersects the ray at point A. The teacher then asked him as to why he did what he did, she started from the back and asked him as to how the intersection of the ray and the perpendicular bisector gives A.

Which of the following is the reason for him drawing the perpendicular bisector and intersecting it with the ray?