Cube is the total surface area. How to find the area of a cube
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The cube has many interesting mathematical properties and has been known to people since ancient times. Representatives of some ancient Greek schools believed that elementary particles(atoms) that make up our world are cube-shaped, and mystics and esotericists even deified this figure. And today, representatives of parascience attribute amazing energy properties to the cube.
The cube is an ideal figure, one of the five Platonic solids. The Platonic solid is
a regular polyhedral figure that satisfies three conditions:
1. All its edges and faces are equal.
2. The angles between the faces are equal (for a cube, the angles between the faces are equal and amount to 90 degrees).
3. All vertices of the figure touch the surface of the sphere described around it.
The exact number of these figures was named by the ancient Greek mathematician Theaetetus of Athens, and Plato’s student Euclid in the 13th book of the Elements gave them a detailed mathematical description.
The ancient Greeks, inclined to use quantitative values to describe the structure of our world, gave the Platonic solids a deep sacred meaning. They believed that each of the figures symbolizes universal principles: tetrahedron - fire, cube - earth, octahedron - air, icosahedron - water, dodecahedron - ether. The sphere described around them symbolized perfection, the divine principle.
So, a cube, also called a hexahedron (from the Greek “hex” - 6), is a three-dimensional regular It is also called a rectangular parallelepiped.
A cube has six faces, twelve edges and eight vertices. Other tetrahedrons (tetrahedron with triangle-shaped faces), octahedron (octahedron) and icosahedron (twenty-hedron) can be inscribed into this figure.
It is called a segment connecting two vertices that are symmetrical relative to the center. Knowing the length of the cube edge a, you can find the length of the diagonal v: v = a 3.
As mentioned above, a sphere can be inscribed into a cube, and the radius of the inscribed sphere (denoted by r) will be equal to half the length of the edge: r = (1/2)a.
If a sphere is described around a cube, then the radius of the described sphere (let’s denote it R) will be equal to: R= (3/2)a.
A fairly common question in school problems: how to calculate area
cube surface? It’s very simple, just visualize a cube. The surface of the cube consists of six square-shaped faces. Therefore, in order to find the surface area of a cube, you first need to find the area of one of the faces and multiply by their number: S p = 6a 2.
In the same way as we found the surface area of a cube, let’s calculate the area of its side faces: S b =4a 2.
From this formula it is clear that the two opposite faces of the cube are the bases, and the remaining four are the side surfaces.
You can find the cube in another way. Considering the fact that a cube is a rectangular parallelepiped, we can use the concept of three spatial dimensions. This means that the cube, being a three-dimensional figure, has 3 parameters: length (a), width (b) and height (c).
Using these parameters, we calculate the total surface area of the cube: S p = 2(ab+ac+bc).
The volume of a cube is the product of three components - height, length and width:
V= abc or three adjacent edges: V=a 3.
A cube is one of the simplest three-dimensional figures. Everyone is familiar with ice cubes, square boxes or salt crystals - they are all such shapes. The surface area of a cube is the total area of all sides on its surface. All six of its faces are proportional, therefore, knowing the length of one of them, you can calculate the lateral area and surface area of any figure.
How to find the area of a cube - what does the figure represent?
A cube is a three-dimensional figure that has the same dimensions. Its length, width and height are identical, and each edge meets the other edges at the same angle. Finding the surface area of a cube is quick and convenient because it is made up of congruent or commensurate squares. So, once you find the size of one of the squares, you will know the area of the entire shape.
How to find the area of a cube - the faces of the figure
From the illustration it can be seen that the cube has a front and a back face, two sides and a top and bottom side. The area of any cube will be six congruent squares. In fact, if you unfold it, you can clearly see the six squares that make up the overall surface of the figure.
How to find the area of a cube
The area of a cube consists of the area of its six faces. Since they are all equal, it is enough to know the area of one of them and multiply the value by 6. The area of the figure is also found using a simple formula: S = 6 x a², where “a” is one of the sides of the cube.
How to find the area of a cube - find the area of the side
- Let's assume that the height of the cube is 2 cm. Since its surface is made of squares, all its edges will be the same length. Therefore, based on the height dimensions, its length and width will be 2 cm.
- To find the area of one of the squares, remember your basic knowledge of geometry, where S = a², where a is the length of one of the sides. In our case, a = 2 cm, so S = (2 cm)² = 2 cm x 2 cm = 4 cm².
- The area of one of the surface squares is 4 cm². Don't forget to include your value in square units.
How to find the area of a cube - example
Since the entire surface of the figure consists of six proportionate squares, you need to multiply the area of one side by 6, following the formula S = 6 x a². In our case, S = 6 x 4 cm² = 24 cm². The area of the three-dimensional figure is 24 cm².
Find the area of a cube if the side is expressed in fractions
If you have trouble working with fractions, convert them to a decimal.
For example, the height of a cube is 2 ½ cm.
- S = 6 x (2½ cm)²
- S = 6 x (2.5 cm)²
- S = 6 x 6.25 cm²
- S = 37.5 cm²
- The surface area of the cube is 37.5 cm².
Knowing the area of the cube, we find its side
If the surface area of a cube is known, the length of its sides can be determined.
- The area of the cube is 86.64 cm². It is necessary to determine the length of the edge.
- Solution. Since the surface area is known, you need to count backwards, divide the value by 6, and then take the square root.
- Having made the necessary calculations, we obtain a length of 3.8 cm.
How to find the area of a cube - online area measurement
Using the calculator on the OnlineMSchool website, you can quickly calculate the area of a cube. It is enough to enter the desired side value and the service will provide a detailed step-by-step solution to the task.
So, to know the area of a cube, calculate the area of one of the sides, then multiply the result by 6, since the figure has 6 equal sides. When calculating, you can use the formula S = 6a². If the surface area is given, it is possible to determine the side length by working backwards.
Geometry is one of the basic mathematical sciences, the basic course of which is studied even at school. In fact, the benefits of knowing various figures and laws will be useful to everyone in life. Very often there are geometric problems on finding area. If with flat figures students don’t have any special problems, so volumetric may cause some difficulties. Calculate cube surface area It's not as simple as it seems at first glance. But with due attention, even the most difficult task can be solved.
Necessary:
Knowledge of basic formulas;
- conditions of the problem.
Instructions:
- First of all, you need to decide which formula for the area of a cube is applicable in a particular case. To do this you need to look at given parameters of the figure . What data is known: rib length, volume, diagonal, face area. Depending on this, the formula is selected.
- If according to the conditions of the problem it is known cube edge length, then it is enough to apply the simplest formula to find the area. Almost everyone knows that the area of a square is found by multiplying the lengths of its two sides. Cube faces- squares, therefore, its surface area is equal to the sum of the areas of these squares. A cube has six sides, so the formula for the area of a cube would look like this: S=6*x 2 . Where X - cube edge length.
- Let's assume that cube edge not specified, but known. Since the volume of a given figure is calculated by raising it to the third power the length of his rib, then the latter can be obtained quite easily. To do this, it is necessary to extract the third root from the number indicating the volume. For example, for a number 27 the third root of the number is 3 . Well, we’ve already discussed what to do next. Thus, the formula for the area of a cube with a known volume also exists, where instead of X is the third root of the volume.
- It happens that it is only known diagonal length . If you remember Pythagorean theorem, then the length of the edge can be easily calculated. There is enough basic knowledge here. The result obtained is substituted into the formula for the surface area of a cube that we already know: S=6*x 2 .
- To summarize, it is worth noting that for correct calculations you need to know the length of the edge. The conditions in the tasks are very different, so you should learn to perform several actions at once. If other characteristics are known geometric figure, then using additional formulas and theorems you can calculate the edge of the cube. And based on the result obtained, calculate the result.
By cube we mean a regular polyhedron, all of whose faces are formed by regular quadrangles - squares. Finding the area of the face of any cube does not require heavy calculations.
Instructions
To begin with, it is worth focusing on the very definition of a cube. It shows that any of the faces of the cube is a square. Thus, the task of finding the area of a cube face is reduced to the task of finding the area of any of the squares (cube faces). You can take exactly any of the faces of the cube, since the lengths of all its edges are equal to each other.
In order to find the area of the face of a cube, you need to multiply any pair of its sides, because they are all equal to each other. This can be expressed by the formula:
S = a?, where a is the side of the square (edge of the cube).
Example: The length of an edge of a cube is 11 cm; you need to find its area.
Solution: knowing the length of the face, you can find its area:
S = 11? = 121 cm?
Answer: The area of the face of a cube with an edge of 11 cm is equal to 121 cm?
Please note
Any cube has 8 vertices, 12 edges, 6 faces and 3 vertex faces.
A cube is a figure that is found incredibly often in everyday life. Suffice it to recall game cubes, dice, cubes in various children's and teenage construction sets.
Many architectural elements are cubic in shape.
Cubic meters are used to measure the volume of various substances in various fields life of society.
Scientifically speaking, a cubic meter is a measure of the volume of a substance that can fit into a cube with an edge length of 1 m
Thus, you can enter other units of volume measurement: cubic millimeters, centimeters, decimeters, etc.
In addition to various cubic units of volume measurement, in the oil and gas industry it is possible to use another unit - the barrel (1m? = 6.29 barrels)
Useful advice
If the length of its edge is known for a cube, then, in addition to the area of the face, you can find other parameters of this cube, for example:
Surface area of the cube: S = 6*a?;
Volume: V = 6*a?;
Radius of the inscribed sphere: r = a/2;
Radius of a sphere circumscribed around a cube: R = ((?3)*a))/2;
Diagonal of a cube (a segment connecting two opposite vertices of a cube that passes through its center): d = a*?3
This is the total area of all surfaces of the figure. The surface area of a cube is equal to the sum of the areas of all its six faces. Surface area is a numerical characteristic of a surface. To calculate the surface area of a cube, you need to know a certain formula and the length of one of the sides of the cube. In order for you to quickly calculate the surface area of a cube, you need to remember the formula and the procedure itself. Below we will discuss in detail the calculation procedure. total surface area of the cube and give specific examples.
Performed according to the formula SA = 6a 2. Cube (regular hexahedron) is one of 5 types regular polyhedra, which is a regular cuboid, the cube has 6 faces, each of these faces is a square.
For calculating the surface area of a cube You need to write down the formula SA = 6a 2. Now let's look at why this formula looks like this. As we said earlier, a cube has six equal square faces. Based on the fact that the sides of the square are equal, the area of the square is - a 2, where a is the side of the cube. Since a cube has 6 equal square faces, then to determine its surface area, you need to multiply the area of one face (square) by six. As a result, we obtain a formula for calculating the surface area (SA) of a cube: SA = 6a 2, where a is the edge of the cube (side of the square).
What is the surface area of a cube?
It is measured in square units, for example, mm 2, cm 2, m 2 and so on. For further calculations you will need to measure the edge of the cube. As we know, the edges of a cube are equal, so it will be enough for you to measure only one (any) edge of the cube. You can perform this measurement using a ruler (or tape measure). Pay attention to the units of measurement on the ruler or tape measure and write down the value, denoting it with a.
Example: a = 2 cm.
Square the resulting value. Thus, you square the length of the edge of the cube. To square a number, multiply it by itself. Our formula will look like this: SA = 6*a 2
You have calculated the area of one of the faces of a cube.
Example: a = 2 cm
a 2 = 2 x 2 = 4 cm 2
Multiply the resulting value by six. Don't forget that a cube has 6 equal sides. Having determined the area of one of the faces, multiply the resulting value by 6 so that all faces of the cube are included in the calculation.
Here we come to the final action calculating the surface area of a cube.
Example: a 2 = 4 cm 2
SA = 6 x a 2 = 6 x 4 = 24 cm 2
