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

Sphere

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

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.

Spherical cap

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

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)$


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