# Sphere

A **sphere** is a solid bounded by a surface all points of which are equally distant a point within, called *center*.

A **radius** is the distance between the center and the surface.

A **diameter** of a sphere is a straight line drawn through the center, having its extremities in the surface.

### Formulas

**Surface area:**

Surface area $=4\pi R^2 = \pi d^2=\sqrt[3]{36\pi V^2}$

**Volume:**

Volume $ =\frac43 \pi R^3 = \frac{\pi}{6}d^3 = \frac{1}{6}\sqrt{\frac{s^3}{\pi}}$

#### Spherical Sector

A **spherical sector** is a portion of a sphere defined by a conical boundary with apex at the center of the sphere.

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

$A = 2 \pi Rh$

Total surface area (including the cone surface):

$A=\pi R(2h + r)$

**Volume:**

Volume $= \frac{2\pi R^2h}{3}$

#### Spherical cap (Spherical segment of one base)

A **spherical cap** is a portion of a sphere cut off by a plane.

The curved surface area of the spherical cap:

Curved surface area $=2\pi Rh = \pi d h=\pi(r^2+h^2)$

Total surface area $=2\pi R h + \pi r^2 = \pi(h^2 + 2r^2) = \pi h(4R - h)$

**Volume:**

Volume $\frac{\pi h^2}{3}(3R - h) = \frac{\pi h}{6}(3r^2 + h^2)$

#### Spherical segment

A **spherical segment** is a portion of the sphere included between two parallel planes.

The curved surface area of the spherical zone - which excludes the top and bottom bases:

Curved surface area $=2\pi R h$

The surface area - which includes the top and bottom bases:

Surface area $=2\pi Rh + \pi r_1^2 + \pi r_2^2 = \pi(2Rh + r_1^2 + r_2^2)$

**Volume:**

Volume $ = \frac{1}{6}\pi h(3r_1^2+3r_2^2+h^2)$