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Mensuration of a Sphere: Surface Area, Volume, Zones



You are given a sphere of radius R units. 
Units may be any units of length: inches, cm, metres, feet, miles, km. 

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

(4/3) * π * Radius3
4 * π * Radius2  




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 π h/6 Cubic Units

Volume =12 3r22 π h/6  Cubic Units

Curved surface area of the zone = 2 π R h 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 units?

As derived here(3R - h)  πh2

Height of the geometric centroid above the centre of the sphere = (3 (2R - h) 2(3R - h) units



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 * Radiusπ * Radius= 3 π Radius



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)

Some 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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