Spherical sector

In geometry, a spherical sector, also known as a spherical cone, is a portion of a ball that is bounded by a spherical cap and the cone that connects the centre of the sphere to the boundary of the cap. It is the three-dimensional analogue of the sector of a circle.

Volume

If the radius of the sphere is denoted by r and the height of the cap by h, the volume of the spherical sector is

V

=

2

π

r

2

h

3

.

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

This may also be written as

V

=

2

π

r

3

3

(

1

−

cos

⁡

φ

)

,

{\displaystyle V={\frac {2\pi r^{3}}{3}}(1-\cos \varphi )\,,}

where φ is half the cone aperture angle, i.e., φ is the angle between the rim of the cap and the axis direction to the middle of the cap as seen from the sphere center. The limiting case is for φ approaching 180 degrees, which then describes a complete sphere.

The height, h is given by

h

=

r

(

1

−

cos

⁡

φ

)

.

{\displaystyle h=r(1-\cos \varphi )\,.}

The volume V of the sector is related to the area A of the cap by:

V

=

r

A

3

.

{\displaystyle V={\frac {rA}{3}}\,.}

Area

The curved surface area of the spherical cap (on the surface of the sphere, excluding the cone surface) is

A

=

2

π

r

h

.

{\displaystyle A=2\pi rh\,.}

It is also

A

=

Ω

r

2

{\displaystyle A=\Omega r^{2}}

where Ω is the solid angle of the spherical sector in steradians, the SI unit of solid angle. One steradian is defined as the solid angle subtended by a cap area of A = r2.