Math
Spherical Sector & Segment Formula  Wedge, Lune, Spherical Cap Volume
Table of Contents
Spherical Cap Volume Formulas
The spherical cap is no different but a part of the sphere itself that falls over the plane of a sphere. If you know the values of the base area, radius, and the height of a sphere then this is easy to calculate the area of the shape. The other popular name for the spherical cap is spherical dome. Sometimes, you will be given the base radius and sometimes you will be given the value of sphere radius.
Make sure that you know the difference between these two terms and apply the respective formula accordingly. Here is given a diagram to give you an accurate idea of the concept. Also, we have given the spherical cap volume formulas to make the thing easier for you.
\[\large V=\frac{\pi}{3}\,H^{2}(3R^{2}=H^{2})\]
Where,
H = height
S = sphere radius
A = base radius
R = sphere radius
Based on the formula given earlier, H will be the height, R is the sphere radius and a is the base radius. The value of pi is fixed in mathematics.
Spherical Sector Formula
The shape of a spherical sector is similar to the ice cream cone. In mathematics, a portion of the sphere with vertices in the middle and the conical boundary will be named as the spherical sector. The base of a spherical sector is termed as the zone as well. The other names for the bases are the union of a cone or spherical cap. The formula of a spherical sector is given as below –
\[\large surface\;Area\;of\;a\;sector\;of\;a\;sphere=\pi r(2h+a)\]
\[\large volume\;of\;a\;sector\;of\;a\;sphere=\frac{2\pi r^{2}h}{3}\]
If you know the height and radius of a spherical sector then calculating the volume of a spherical sector would be easier.
Spherical Segment Formula
A spherical segment is a solid shape defined by cutting the shape into parallel planes. It can be taken as the spherical cap when the top is truncated, so it looks like a spherical frustum too. The base of the spherical segment is named as the zone. At some places, zone, spherical cap, and spherical segment terms are used interchangeably. So, you have to make sure of each of the concept and respective formulas too. The volume of a spherical segment is given as below –
\[\large surface\;Area\;of\;a\;segment\;of\;a\;sphere=2\pi Rh\]
\[\large volume\;of\;a\;segment\;of\;a\;sphere=\frac{\pi h}{6}(3r_{1}^{2}+r_{2}^{2}+h^{2})\]
Where R (r1 + r2) is the radius and ‘h’ is the height of a spherical segment.
Spherical Wedge and Lune Formula
When semicircle is revolved around the diameter with less than 360degree angle then the solid shape will be named as the spherical wedge. At the same time, Spherical Lune is the total space occupied within a sphere and surrounded by two half circles that will meet further at antipodal points. You must be shocked what we meant by antipodal here.
\[\large Volume=\frac{2}{3}\:R^{3}\theta\]
\[\large Surface\;Area=2R^{2}\theta\]
\[\large Arc\;Length\;at\;the\;equator=R\theta\]
This is the point marked on the surface of a sphere which is diametrically opposite to each other. When a line is drawn from one point to another it will make a straight line. Here, we have given the formulas for spherical wedge and Lune to calculate the Volume, surface area, and the arc length.

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