You will need

- Calculator

Instruction

1

Basically, there are four types of tasks that you need to find the edge of the cube. This definition of the length of the cube edges of the square faces of a cube, volume of a cube diagonal of the cube face diagonal of the cube. Consider all four options such tasks. (The rest of the task tend to be variations of the above or a task in trigonometry, having a very indirect relation to the subject)

If famous square faces of a cube, find the edge of the cube is very simple. As the face of the cube is a square with a side equal to the edge of the cube, its area equals the square of the cube. Hence the edge length of the cube equals the square root of the square of its faces, ie:

a=√S, where

a - the edge length of the cube

S - square face of the cube.

If famous square faces of a cube, find the edge of the cube is very simple. As the face of the cube is a square with a side equal to the edge of the cube, its area equals the square of the cube. Hence the edge length of the cube equals the square root of the square of its faces, ie:

a=√S, where

a - the edge length of the cube

S - square face of the cube.

2

Finding the face of the cube by its volume even easier. Given that the volume of a cube equals the cube (third-degree) length of the cube, we get that the edge length of the cube equals the cube root (third-degree) from its scope, ie:

a=√V (cube root), where

a - the edge length of the cube

V is the volume of the cube.

a=√V (cube root), where

a - the edge length of the cube

V is the volume of the cube.

3

A little harder to find the length of the edges of the cube according to the known lengths of the diagonals. We denote by:

a - the edge length of the cube;

b - the length of the diagonal of the cube face;

c - the length of the diagonal of the cube.

As can be seen from the figure, the diagonal face and edges of the cube form a rectangular equilateral triangle. Therefore, by the Pythagorean theorem:

a^2+a^2=b^2

(^ - icon exponentiation).

Hence, we find:

a=√(b^2/2)

(to find the edge of the cube to extract the square root of half the square of the diagonal faces).

a - the edge length of the cube;

b - the length of the diagonal of the cube face;

c - the length of the diagonal of the cube.

As can be seen from the figure, the diagonal face and edges of the cube form a rectangular equilateral triangle. Therefore, by the Pythagorean theorem:

a^2+a^2=b^2

(^ - icon exponentiation).

Hence, we find:

a=√(b^2/2)

(to find the edge of the cube to extract the square root of half the square of the diagonal faces).

4

To find the edge of the cube its diagonal, again use a picture. The diagonal of a cube (C) diagonal edge (b) and the edge of the cube (a) form a rectangular triangle. So according to the Pythagorean theorem:

a^2+b^2=c^2.

Use vyshesteblievskoy dependence between a and b and substitute into the formula

b^2=a^2+a^2. Received:

a^2+a^2+a^2=c^2, where we find:

3*a^2=c^2, therefore:

a=√(c^2/3).

a^2+b^2=c^2.

Use vyshesteblievskoy dependence between a and b and substitute into the formula

b^2=a^2+a^2. Received:

a^2+a^2+a^2=c^2, where we find:

3*a^2=c^2, therefore:

a=√(c^2/3).

# Advice 2: How to find the area and volume of a cube

A cube is a rectangular parallelepiped of which all edges are equal. Therefore, the General formula for the volume of the rectangular prism and the formula for its surface area in the case of

**Cuba**easier. The volume**of a cube**and**the area of**the surface can be found, knowing the volume of a sphere inscribed in it, or sphere, described around it.You will need

- the length of a side of a cube, the radius of the inscribed and circumscribed ball

Instruction

1

The volume of a box is: V = abc, where a, b, c is its dimension. Therefore, the volume

**of a cube**is equal to V = a*a*a = a^3, where a is the side length**of the cube**.The surface area of**a cube**is equal to the sum of the areas of all its faces. Only**the cube**has six faces, so**the area of**its surface equal to S = 6*(a^2).2

Let the ball inscribed in the cube. Obviously, the diameter of the ball is equal to the side

**of the cube**. Substituting the length of the diameter in the expression for the volume instead of the length of the edges**of the cube**and using that the diameter is twice the radius, then get V = d*d*d = 2r*2r*2r = 8*(r^3) where d is the diameter of the inscribed circle, and r is the radius of the inscribed circle.The surface area of**the cube**will then be equal to S = 6*(d^2) = 24*(r^2).3

Let the ball circumscribed around

Consider first one of the faces

**the cube**. Then its diameter will coincide with the diagonal**of the cube**. The diagonal**of the cube**passes through the center**of the cube**and connects two opposite points.Consider first one of the faces

**of the cube**. The edges of this face are the legs of a right triangle in which the face diagonal d is the hypotenuse. Then by the Pythagorean theorem we get: d = sqrt((a^2)+(a^2)) = sqrt(2)*a.4

Then consider the triangle in which the hypotenuse is the diagonal

So, derived the formula of diagonal

**of a cube**and the diagonal edge of d and one edge**of cube**a to his legs. Similarly, by the Pythagorean theorem we get: D = sqrt((d^2)+(a^2)) = sqrt(2*(a^2)+(a^2)) = a*sqrt(3).So, derived the formula of diagonal

**of a cube**is equal to D = a*sqrt(3). Hence, a = D/sqrt(3) = 2R/sqrt(3). Therefore, V = 8*(R^3)/(3*sqrt(3)), where R is the radius of the ball is described.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).