Volume of Sphere Formula Volume of Sphere Formula A sphere is a rounded geometrical object in 3 – D like the shape of a round ball. A circle is a two dimensional object while a sphere is completely symmetrical around its center having all points on the surface lying at the same distance ‘r’ from the center point. The maximum straight distance through the sphere is named as the diameter of the sphere that passes through the center. The radius of a circle and a sphere is half to the diameter. In a three dimensional space the volume of a sphere is given by the following expression, V = (4/3) ?r3, Here ‘r’ shows the radius of the sphere and ‘?’ is a constant. The formula first derived by the mathematician Archimedes according to him the volume of a sphere is 2/3 that of a circumscribed cylinder. The Volume of a Sphere Formula can be derived as below.
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The incremental volume ∂V at any given point ‘x’ by the product of the cross sectional area of the disk at the point ‘x’ and the thickness is ∂x, then the volume is, ∂V = ?y2 . ∂x, The total volume of the sphere is the summation of all incremental values, thus V = ∑ ?y2 . ∂x, In the limit as the thickness ∂x approaches to zero the volume becomes V = -r∫r ?y2 . dx, At any given point on the sphere a right angled triangle connects all three x, y and r to the origin hence from the Pythagorean theorem, r2 = x2 + y2, Thus on substituting ‘y’ with a function of ‘x’, that gives V = -r∫r ? (r2 - x2) . dx, This can be solved as, V = ? [r2x – x3 / 3]-rr, V = ? (r3 – r3 / 3) - ? (-r3 + r3 / 3), V = (4/3) ?r3, Therefore the Volume of a Sphere Formula is given as
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V = (4/3) ?r3, This formula is useful in many cases when there is a need to find the volume related to the balls or spheres. Alternatively there is an another formula found by using spherical co ordinates with the volume element dV = r2 sinθ dr dθ d?, The volume of a sphere can be calculated by the below three methods, If a sphere consists of an infinite number of infinitesimally thick spherical shells then the volume of the sphere could be determined by the integration that is summing up the volumes of such shells. The surface area of a sphere having the radius r is “4?r2” then the volume of a spherical shell of radius r and the thickness ‘dr’ is given as, dV = 4?r2 dr, If a sphere consists of an infinite number of infinitesimally thick circular plane parallel slices then the volume of the sphere is calculated by summing up the volumes of such slices. The area of a circular disk of radius r is ‘?r2’ then the volume of the disk having thickness ‘dz’ is dV = ?r2 dz, Since Z2 = R2 – r2, So dV = ? (R2 – Z2) dz, If a sphere consists of an infinite number of infinitesimally thick centric cylindrical shells of variable length then the volume of the sphere would be the summation of the volumes of such shells. The volume of the infinitesimal shell is then given as addsplace.co.cc dV = -2 ? rZ dr, Since we know that, Z2 = R2 – r2, ZdZ = -rdr, So dV = 2? Z2dZ.
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