The volume of a geometrical figure - one of its parameters which is quantitatively characterizing space which this figure occupies. Volume figures have also other parameter - surface area. These two indicators are connected among themselves by certain ratios what allows, in particular? to calculate the volume of the correct figures, knowing the area of their surface.
Instruction
1. The surface area of the sphere (S) can be expressed as the quadruple work of number of Pi on the squared radius (R): S = 4*π*R². The volume (V) sphere limited to this sphere can be expressed through radius too - it is directly proportional to the work of quadruple number of Pi on the radius cubed and is inversely proportional to the three: V = 4*π*R³/3. Use these two expressions to receive a formula of calculation of volume, having connected them through radius - express radius from the first equality (= ½ * √ (S/π) and set up R him in the second identity: V = 4*π * (½ * √ (S/π)³/3 = ⅙ *π * (√ (S/π)³.
2. The similar couple of expressions can be made for surface area (S) and the volume (V) cube, having connected them through length of an edge (a) of this polyhedron. Volume is equal to the third degree of length of an edge (√ = a³), and surface area - the second degree of the same parameter of a figure increased six times (V = 6*a²). Express edge length through surface area (a =³ √ V) and substitute in a formula of calculation of volume: V = 6 * ³ √ V)².
3. The volume of the sphere (V) can be calculated also on the area not of a full surface, and only separate segment (s) which height (h) is known too. The area of such site of a surface has to be equal to the work of the doubled Pi's number on the radius of the sphere (R) and height of a segment: s = 2*π*R*h. Find from this equality radius (R = s / (2*π*h)) and substitute in the formula connecting volume with radius (V = 4*π*R³/3). As a result of simplification of a formula at you such expression has to turn out: V = 4*π * (s / (2*π*h))³/3 = 4*π*s³ / (8*π³*h³)/3 = s³ / (6*π²*h³).
4. For calculation of volume of a cube (V) on the area of one its side (s) of any additional parameters the nobility is not required. Length of an edge (a) of the correct hexahedron can be found extraction of a square root from the area of a side (a = √s). Substitute this expression in the formula connecting volume with the cube edge size (V = a³): V = (√s)³.