### 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. 332 2 Great Circles and Small CirclesA 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 r22Volume of a zone or frustum =   (3r12 3r22 2 π h/6 Cubic UnitsConsider 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 unitsAgain applying the Pythagoras Theorem,  r2= √(R222 = √( 11.022 unitsh = h12 Volume =12 3r22 2 π h/6 = (3 * 10.9523 * 10.8222π * 3/6  Cubic Units = 1130.97  Cubic UnitsCurved surface area of the zone = 2 π R h = 207.35 Square UnitsArea of the upper base = π r12Square UnitsArea of the lower base = π r22 Square UnitsTotal Surface Area = 2 π R h +  π r12 π r22 Square UnitsMensuration for a Spherical Cap What is the volume of a spherical cap of height h = 3 units?As derived here(3R - h)  πh2Height 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 unitsWhat 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 unitsSome 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