area of spherical zone

Let us consider the circle


with radius r and centre  (r, 0).  A spherical zone may be thought to be formed when an arc of the circle rotates around the x-axis.  For finding the are of the zone, we can use the formula

A= 2πaby1+(dydx)2𝑑x (1)

of the entry area of surface of revolution.  Let the ends of the arc correspond the values a and b of the abscissaMathworldPlanetmath such that  b-a=h  is the of the spherical zone.  In the formula, we must use the solved form


of the equation of the circle.  The formula then yields

A= 2πabrx-x21+(r-xrx-x2)2𝑑x= 2πabr𝑑x= 2πr(b-a).

Hence the area of a spherical zone (and also of a spherical calotte) is

A= 2πrh. (2)

From here one obtains as a special case  h=2r  the area of the whole sphere:

A= 4πr2. (3)

Remark.  The formula (2) implies that the centre of mass of a half-sphere is at the halfway point of the axis of symmetryMathworldPlanetmath (h=r2).

Title area of spherical zone
Canonical name AreaOfSphericalZone
Date of creation 2013-03-22 18:19:05
Last modified on 2013-03-22 18:19:05
Owner pahio (2872)
Last modified by pahio (2872)
Numerical id 9
Author pahio (2872)
Entry type Derivation
Classification msc 26B15
Classification msc 53A05
Classification msc 51M04
Synonym area of spherical calotte
Related topic AreaOfTheNSphere
Related topic CentreOfMassOfHalfDisc