If six sides of square shape limit some volume of space, then it is possible to call a geometrical form of this space cubic or hexahedral. All twelve edges of such spatial figure have identical length that considerably simplifies calculations of parameters of a polyhedron. Cube diagonal length - not an exception, it is possible to find it in many ways.
Instruction
1. If length of an edge of a cube (a) is known from statements of the problem, the formula of calculation of length of diagonal of a side (l) can be brought out of Pythagorean theorem. Cubed any two adjacent edges form a right angle therefore the triangle made of them and diagonals of a side, is rectangular. Edges in this case are legs, and it is necessary to calculate to you hypotenuse length. According to the theorem mentioned above it is equal to a square root from the sum of squares of lengths of legs and as in this case they have the identical sizes, just increase edge length by a square root from the two: l = √ (a²+a²) = √ (2*a²) = a * √ 2.
2. The area of a square can be expressed too through the diagonal length and as each side of a cube has such form, knowledge of the area of a side (s) is enough for calculation of its diagonal (l). The area of each side surface of a cube is equal to the squared edge length therefore the party of a square of a side can be expressed through it as √s. Substitute this value in a formula from the previous step: l = √s * √ 2 = √ (2*s).
3. The cube is made of six sides of an identical form therefore if in statements of the problem the total area of a surface (S) is given, for calculation of diagonal of a side (l) to change rather a little a formula of the previous step. Replace in it the area of one side of the one sixth total area: l = √ (2*S/6) = √ (S/3).
4. Length of an edge of a cube can be expressed also through the volume of this figure (V), and it allows to use a formula of calculation of length of diagonal of a side (l) from the first step and in this case, having made to it some amendments. The volume of such polyhedron is equal thirds of degree of length of an edge therefore replace in a formula length of the party of a side with a cubic root from volume: l =³ √ V * √ 2.
5. Radius of the sphere (R) described about a cube is connected with edge length in the coefficient equal to a half of a root of the three. Express the party of a side through this radius and substitute expression in the same formula of calculation of length of diagonal of a side from the first step: l = R*2 / √ 3 * √ 2 = R * √ 8 / √ 3.
6. The formula of calculation of diagonal of a side (l) with use of radius of the sphere (r) entered in a cube will be even simpler as this radius is a half of length of an edge: l = 2*r * √ 2 = r * √ 8.