How To Find A Diagonal Of The Cube | The Science

 

 

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How to find a diagonal of the cube

If the six faces of a square shape some limited amount of space, the geometric shape of this space can be called cubic or hexahedral. All twelve edges of such spatial figures are the same length, which greatly simplifies the calculation of the parameters of the polyhedron. The length of the diagonal of the cube - not the exception, it can be found in many ways.

How to find a diagonal of the cube

Instruction how to find the diagonal of the cube

Step 1:

If the length of the cube edge (a) of the conditions of the problem is known, the formula for calculating the length of the diagonal of the face (l) can be derived from the Pythagorean theorem. In Cuba, any two adjacent edges to form a right angle, so the triangle made of them and the diagonal face is rectangular. The ribs in this case - legs are, and you need to calculate the length of the hypotenuse. According to the above-mentioned theorem, it is equal to the square root of the sum of the squares of the lengths of the legs, as well as in this case they are the same size, simply multiply the edge length to the square root of twos: l = √ (a² + a²) = √ (2 * a²) = a * √2.

Step 2:

The area of ​​the square can also be expressed in terms of the length of the diagonal, and as each face of the cube is just such a shape, the face area of ​​knowledge (s) is sufficient to calculate its diagonal (l). The area of ​​each side of the cube is equal to the squared length of the ribs, so the direction of the square faces can be expressed in terms of how it √s. Substitute this value into the formula from the previous step: l = √s * √2 = √ (2 * s).

Step 3:

The cube is made up of the six faces of the same shape, so if in the conditions of the task given to the total surface area (S), to calculate the diagonal of the face (l) is enough to slightly change the formula of the previous step. Replace it faces one area of ​​one-sixth of the total area: l = √ (2 * S / 6) = √ (S / 3).

Step 4:

The length of the edges of the cube can be expressed through the bulk of this figure (V), and it allows the formula for calculating the length of the diagonal of the face (l) of the first steps to use and in this case, by making some amendments. The volume of this polyhedron is equal to the third degree of edge length, so replace the formula in the length of the side faces of the cube root of the volume: l = ³√V * √2.

Step 5:

The radius of the sphere circumscribed around the cube (R) is associated with the edge length ratio equal to half the square root of three. Give the side faces through this range and substitute the expression all the same formula for calculating the length of the diagonal of the face of the first step: l = R * 2 / √3 * √2 = R * √8 / √3.

Step 6:

The formula for calculating the diagonal faces (l) using the radius of the sphere inscribed in a cube (r) will be even easier, since the radius is half the edge length: l = 2 * r * √2 = r * √8.