Ball wedge

A spherical wedge with radius r and opening angle α

In terms of geometry, a spherical wedge is a part of a solid sphere that is cut out by two half-planes intersecting one another in one diameter . The edge (the surface ) consists of a spherical triangle and two semicircles with the radius of the sphere.

If you consider the full sphere (radius r) as a body of revolution that is created by rotating a semicircle around the limiting diameter by an angle of 360 °, a spherical wedge is created if the semicircle is only rotated by an angle (the "opening angle" of the wedge) . ${\displaystyle \alpha <360^{\circ }}$

The volume of the ball wedge is intuitively proportional to its opening angle and therefore results from

${\displaystyle V={\frac {\alpha }{360^{\circ }}}\cdot V_{Kugel}={\frac {\alpha }{360^{\circ }}}\cdot {\frac {4\pi }{3}}r^{3}}$.

Because of this, this can be simplified ${\displaystyle 360^{\circ }=2\pi }$

${\displaystyle V={\frac {\alpha }{2\pi }}\cdot {\frac {4\pi }{3}}r^{3}={\frac {2}{3}}\alpha r^{3}}$.

Analogously one obtains for the area of the spherical triangle

${\displaystyle A={\frac {\alpha }{360^{\circ }}}\cdot O_{Kugel}={\frac {\alpha }{360^{\circ }}}\cdot 4\pi r^{2}=2\alpha r^{2}}$.

The following applies to the surface of the ball wedge

${\displaystyle O=A+\pi r^{2}=(2\alpha +\pi )r^{2}}$.

The formulas for the volume and the area of ​​the spherical triangle can be derived exactly with the aid of spherical coordinates and volume or area integrals .

More ball parts

• Sphere cutout
• Spherical segment
• Spherical layer
• Ball ring

literature

• Small encyclopedia of mathematics. Verlag Harri Deutsch (1977), ISBN 3871443239 , p. 214.

Web links

• Eric W. Weisstein : Spherical Wedge . In: MathWorld (English).