PlanetMath: area of spherical zone

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area of spherical zone

(Derivation)

Let us consider the circle $$(x-r)^2+y^2 = r^2$$ 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

$\displaystyle A = 2\pi\!\int_{a}^{b}y\sqrt{1+\left(\frac{dy}{dx}\right)^2}\,dx$

(1)

of the entry area of surface of revolution. Let the ends of the arc correspond the values $a$ and $b$ of the abscissa such that $b\!-\!a = h$ is the height of the spherical zone. In the formula, we must use the solved form $$y = (\pm)\sqrt{rx-x^2}$$ of the equation of the circle. The formula then yields $$A = 2\pi\!\int_a^b\sqrt{rx-x^2}\,\sqrt{1+\left(\frac{r-x}{\sqrt{rx-x^2}}\right)^2}\,dx = 2\pi\!\int_a^br\,dx = 2\pi r(b\!-\!a).$$ Hence the area of a spherical zone (and also of a spherical calotte) is

$\displaystyle A = 2\pi rh.$

(2)

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

$\displaystyle A = 4\pi r^2.$

(3)

"area of spherical zone" is owned by pahio.

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See Also: area of the $n$

-sphere

Other names:

area of spherical calotte

Keywords:

area of sphere

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Attachments:: area of spherical calotte by means of chord (Derivation) by pahio

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Cross-references: sphere, spherical calotte, area, equation, abscissa, area of surface of revolution, rotates, arc, centre, radius, circle
There are 2 references to this entry.

This is version 4 of area of spherical zone, born on 2008-08-15, modified 2008-08-18.
Object id is 10944, canonical name is AreaOfSphericalZone.
Accessed 1273 times total.

Classification:

AMS MSC:	51M04 (Geometry :: Real and complex geometry :: Elementary problems in Euclidean geometries)
	53A05 (Differential geometry :: Classical differential geometry :: Surfaces in Euclidean space)
	26B15 (Real functions :: Functions of several variables :: Integration: length, area, volume)

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