The cube is a rectangular parallelepiped which all edges are equal. Therefore the general formula for the volume of a rectangular parallelepiped and a formula for the area of its surface in case of a cube become simpler. Also the volume of a cube and its surface area can be found, knowing the volume of the sphere entered in it or the sphere described around it.
It is required to you
- length of the party of a cube, radius of the entered and described sphere
Instruction
1. The volume of a rectangular parallelepiped is equal: V = to ABC - where a, b, c - its measurements. Therefore the volume of a cube is equal to V = to a*a*a = to a^3 where an is length of the party of a cube. The surface area of a cube is equal to the sum of the areas of all its sides. In total at a cube six sides therefore the area of its surface is equal to S = 6 * (a^2).
2. Let the sphere be entered in a cube. Obviously, diameter of this sphere will be equal to the party of a cube. Substituting diameter length expressions for volume instead of length of an edge of a cube and using that diameter is equal to the doubled radius, we will receive then V = d*d*d = 2r*2r*2r = 8 * (r^3) where d is diameter of an inscribed circle, and r is the radius of an inscribed circle. Cube surface area then will be equal to S = 6 * (d^2) = 24 * (r^2).
3. Let the sphere be described around a cube. Then its diameter will coincide with cube diagonal. Diagonal of a cube passes through the center of a cube and connects two of its opposite points. Consider for a start one of cube sides. Edges of this side are legs of a rectangular triangle in which the diagonal of a side of d will be a hypotenuse. Then on Pythagorean theorem we will receive: d = sqrt ((a^2)+ (a^2)) = sqrt (2) *a.
4. Then consider a triangle in which the cube diagonal, and diagonal of a side of d and one of a cube edges - its legs will be a hypotenuse. Similarly, on Pythagorean theorem we will receive: D = sqrt ((d^2)+ (a^2)) = sqrt (2 * (a^2)+ (a^2)) = a*sqrt(3). So, on the removed formula the diagonal of a cube is equal to D = to a*sqrt (3). From here, a = D/sqrt(3) = 2R/sqrt (3). Therefore, V = 8*(R^3) / (3*sqrt (3)) where R is the radius of the described sphere. The surface area of a cube is equal to S = 6 * (D/sqrt (3)) ^2) = 6*(D^2)/3 = 2*(D^2) = 8*(R^2).