**protection click fraud**

O **cube**, also known as a hexahedron, is a geometric solid which has six faces, all of them made up of squares. In addition to the 6 faces, the cube has 12 edges and 8 vertices. studied in Spatial Geometry, the cube has all its edges congruent and perpendicular, so it is classified as a regular polyhedron. We can perceive the presence of the cube format in our daily lives, in common data used in games, packaging, boxes, among other objects.

**Read too: **Pyramid — geometric solid that has all its faces formed by triangles

**cube summary**

The cube is also known as a hexahedron, because it has 6 faces.

The cube is composed of 6 faces, 12 edges and 8 vertices.

The cube has all its faces formed by squares, so its edges are congruent, and therefore it is a regular polyhedron, also known as Plato's solid.

The area of the base of the cube is equal to the area of a square. Being

*The*the measure of the edge, to calculate the area of the base, we have that:

\(A_b=a^2\)

The lateral area of the cube is formed by 4 squares of sides measuring

*The*, so to calculate it, we use the formula:

\(A_l=4a^2\)

To calculate the total area of the cube, just add the area of its two bases with the lateral area. So, we use the formula:

\(A_T=6a^2\)

The volume of the cube is calculated by the formula:

*\(V=a^3\)*

The measure of the side diagonal of the cube is calculated by the formula:

*\(b=a\sqrt2\)*

The measure of the diagonal of the cube is calculated by the formula:

*\(d=a\sqrt3\)*

**What is cube? **

The cube is a geometric solid composed of 12 edges, 8 vertices and 6 faces. Due to the fact that it has 6 faces, the cube is also known as a hexahedron.

**Cube Composition Elements**

Knowing that the cube has 12 edges, 8 vertices and 6 faces, see the following image.

A, B, C, D, E, F, G and H are the vertices of the cube.

\(\overline{AB},\ \overline{AD},\ \overline{AE},\ \overline{BC},\ \overline{BF},\ \overline{CD,\ }\overline{CG}, \ \overline{DH,\ }\overline{HG},\ \overline{EH}\overline{,\ EF},\ \overline{FG}\) are the edges of the cube.

ABCD, ABFE, BCFG, EFGH, ADHE, CDHG are the faces of the cube.

The cube is composed of 6 square faces, so all its edges are congruent. Because its edges have the same measure, the cube is classified as a polyhedron Plato's regular or solid, together with the tetrahedron, octahedron, icosahedron and dodecahedron.

**cube planning**

To calculate the cube area, it is important to analyze your planning. The cube's unfolding is composed of 6 squares, all congruent with each other:

The cube is made up of 2 square bases, and its lateral area is made up of 4 squares, all congruent.

**See too: **Planning of the main geometric solids

**cube formulas**

To calculate the base area, side area, total area and volume of the cube, we will consider the cube with edge measuring *The*.

**Area of the base of a cube**

As the base is formed by a square of edge *The*, the area of the base of the cube is calculated by the formula:

\(A_b=a^2\)

**Example:**

Calculate the measure of the base of a cube that has an edge measuring 12 cm:

**Resolution:**

\(A_b=a^2\)

\(A_b={12}^2\)

\(A_b=144\ cm^2\)

**cube side area**

The side area of the cube is made up of 4 squares, all with sides measuring *The*. Thus, to calculate the lateral area of the cube, the formula is:

\(A_l=4a^2\)

**Example:**

What is the lateral area of a cube that has an edge measuring 8 cm?

**Resolution:**

\(A_l=4a^2\)

\(A_l=4\cdot8^2\)

\(A_l=4\cdot64\)

\(A_l=256\ cm^2\)

**total cube area**

The total area of the cube or simply the area of the cube is the sum area of all cube faces. We know that it has a total of 6 sides, formed by squares of side *The*, then the total area of the cube is calculated by:

\(A_T=6a^2\)

**Example:**

What is the total area of a cube whose edge is 5 cm?

**Resolution:**

\(A_T=6a^2\)

\(A_T=6\cdot5^2\)

\(A_T=6\cdot25\)

\(A_T=150\ cm^2\)

**cube volume**

The volume of a cube is the multiplication the measure of its three dimensions. As they all have the same measure, we have:

*\(V=a^3\)*

**Example:**

What is the volume of a cube that has an edge measuring 7 cm?

**Resolution:**

*\(V=a^3\)*

*\(V=7^3\)*

*\(V=343\ cm^3\)*

**cube diagonals**

On the cube we can draw the side diagonal, that is, the diagonal of its face, and the diagonal of the cube.

#### ◦ **cube side diagonal **

The lateral diagonal or diagonal of a cube face is indicated by the letter *B* in the image. Fur Pythagorean theorem, we have one right triangle of peccaries measuring* The* and hypotenuse measuring *B*:

b² = a² + a²

b² = 2a²

b = \(\sqrt{2a^2}\)

b = \(a\sqrt2\)

Therefore, the formula to calculate the diagonal of a face of the cube is:

*\(b=a\sqrt2\)*

#### ◦ **cube diagonal**

the diagonal *d* of the cube can be calculated using the Pythagorean theorem as well, since we have a right triangle with legs *B*, *The* and hypotenuse measuring *d*:

*\(d^2=a^2+b^2\)*

But we know that b =\(a\sqrt2\):

\(d^2=a^2+\left (a\sqrt2\right)^2\)

\(d^2=a^2+a^2\cdot2\)

\(d^2=a^2+2a^2\)

\(d^2=3a^2\)

\(d=\sqrt{3a^2}\)

*\(d=a\sqrt3\)*

So, to calculate the diagonal of the cube, we use the formula:

*\(d=a\sqrt3\)*

**Know more: **Cylinder — a geometric solid that classifies as a round body

**Cube solved exercises**

**question 1**

The sum of the edges of a cube is 96 cm, so the measure of the total area of this cube is:

A) 64 cm²

B) 128 cm²

C) 232 cm²

D) 256 cm²

E) 384 cm²

**Resolution:**

Alternative E

First, we will calculate the measure of the edge of the cube. Since it has 12 edges and we know that the sum of the 12 edges is 96, we have:

*The* = 96: 12

*The* = 8 cm

Knowing that each edge measures 8 cm, it is now possible to calculate the total area of the cube:

\(A_T=6a^2\)

\(A_T=6\cdot8^2\)

\(A_T=6\cdot64\)

\(A_T=384\ cm^2\)

**question 2**

A water tank needs to be emptied for cleaning. Knowing that it has the shape of a cube with an edge of 2 m and that 70% of this reservoir is already empty, then the volume of this reservoir that is still occupied is:

A) 1.7 m³

B) 2.0 m³

C) 2.4 m³

D) 5.6 m³

E) 8.0 m³

**Resolution:**

Alternative C

First, we will calculate the volume:

*\(V=a^3\)*

*\(V=2^3\)*

*\(V=8\ m^3\)*

If 70% of the volume is empty, then 30% of the volume is occupied. Calculating 30% of 8:

\(0.3\cdot8=2.4\ m^3\)

By Raul Rodrigues de Oliveira

Maths teacher