# Cube: what is it, elements, flattening, formulas

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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

• 1 - Summary about cube
• 2 - What is a cube?
• 3 - Elements of the cube's composition
• 4 - Cube planning
• 5 - Cube formulas
• Area of ​​the base of a cube
• cube side area
• total cube area
• cube volume
• cube diagonals
• 6 - Exercises solved on cube

## 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.

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• 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.

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## 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 equal to 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

Would you like to reference this text in a school or academic work? Look:

OLIVEIRA, Raul Rodrigues de. "Cube"; Brazil School. Available in: https://brasilescola.uol.com.br/matematica/cubo.htm. Accessed on July 23, 2022.

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