Sphere of radius 11 units: Volume,Area,Zones,Caps,Frustum etc


Units may be any units of length: inches, cm, metres, feet, miles, km. So the radius could be any of 11.0 inches, 11.0 cm, 11.0 metres, 11.0 feet, 11.0 miles, 11.0 km, etc. 


A Sphere is a 3D locus of points which are all equidistant from the centre of the sphere.

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2 2 




Great Circles and Small Circles

A great circle is a circular ring on the sphere, the centre for which, coincides with the centre of the sphere. 
So the radius of the Great Circle is same as that for the sphere.

A small circle of a sphere, is a circle drawn on the sphere, with lesser radius than the sphere itself. 



An example of a Zone of a Sphere or Frustum of a Sphere





Total surface area of a zone or frustum = 2 π R h +  π r12π r22
Volume of a zone or frustum =   (3r12 3r22 2 π h/6 Cubic Units

Consider two parallel planes cutting through the sphere. The first one cuts through 1 unit above the centre. The other one cuts though 2 units below the centre. 
What is the volume and surface area of the frustum so formed?

(a) Let's compute the volume.
Applying the Pythagorean Theorem,  r1= √( R212 √( 11.022  10.95 units

Again applying the Pythagoras Theorem,  r2= √(R222 = √( 11.022 units

h = h12

Volume =12 3r22 2 π h/6 = (3 * 10.9523 * 10.8222π * 3/6  Cubic Units = 1130.97  Cubic Units

Curved surface area of the zone = 2 π R h = 207.35 Square Units
Area of the upper base = π r12Square Units
Area of the lower base = π r22 Square Units
Total Surface Area = 2 π R h +  π r12 π r22 Square Units



Mensuration for a Spherical Cap



What is the volume of a spherical cap of height h = 3 units?

As derived here(3R - h)  πh2

Height of the geometric centroid above the centre of the sphere = (3 (2R - h) 24 (3R - h) = 9.03  (substituting h = 3 units and R = 11.0)



Mensuration for a Hemisphere 



Let's cut the sphere into two hemispheres. What is the volume and total surface area, for either of the hemispheres?
Volume of the hemisphere = (2/3) * π * Radius3
Surface Area of the hemisphere = Curved surface area + area of base = 2 * π * Radius2 π * Radius2 = 3 π Radius2 = 1140.4 square units



What is the volume of material used in a sphere of radius 11.0 units a hollow sphere of thickness 2 units?

Inner radius = Outer Radius - Thickness. So the volume of the spherical gap inside = (4/3)π*(Inner Radius)3

In that case, the volume of material required will be  (4/3)π*(11.03 - (11.0-2)3 ) = 2521.65 cubic units


Some more example(s):


Geometric Properties of a sphere which is of radius 12: Properties like Surface Area, Volume and other aspects of mensuration.



Geometric Properties of a sphere which is of radius 13: Properties like Surface Area, Volume and other aspects of mensuration.



properties of a Sphere tutorial over hereCommon CoreGCSE
 


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