The cube is an amazing figure. It is the same on all sides. Any of its faces can instantly become a base or a side. And nothing will change from this. And the formulas for it are always easy to remember. And it doesn’t matter what you need to find - the volume or surface area of the cube. In the latter case, you don’t even need to learn anything new. It is enough to remember only the formula for the area of a square.
What is area?
This quantity is usually denoted Latin letter S. Moreover, this is true for school subjects such as physics and mathematics. It is measured in square units length. Everything depends on the quantities given in the problem. These can be mm, cm, m or km squared. Moreover, there may be cases when the units are not even indicated. We are simply talking about the numerical expression of the area without a name.
So what is area? This is a quantity that is a numerical characteristic of the figure or volumetric body in question. It shows the size of its surface, which is limited by the sides of the figure.
What shape is called a cube?
This figure is a polyhedron. And not easy. It is correct, that is, all its elements are equal to each other. Be it sides or edges. Each surface of the cube is a square.
Another name for a cube is a regular hexahedron, or in Russian, a hexagon. It can be formed from a quadrangular prism or parallelepiped. Subject to the condition that all edges are equal and the angles form 90 degrees.
This figure is so harmonious that it is often used in everyday life. For example, a baby's first toys are blocks. And fun for older ones is the Rubik's Cube.
How is the cube related to other shapes and bodies?
If you draw a section of a cube that passes through its three faces, it will look like a triangle. As you move away from the top, the cross section will become larger. The moment will come when 4 faces will intersect, and the cross-sectional figure will become a quadrilateral. If you draw a section through the center of the cube so that it is perpendicular to its main diagonals, you will get a regular hexagon.
Inside the cube you can draw a tetrahedron ( triangular pyramid). One of its corners is taken as the vertex of the tetrahedron. The remaining three will coincide with the vertices that lie at opposite ends of the edges of the selected corner of the cube.
You can fit an octahedron into it (a convex regular polyhedron that looks like two connected pyramids). To do this, you need to find the centers of all the faces of the cube. They will be the vertices of the octahedron.
The reverse operation is also possible, that is, it is actually possible to fit a cube inside the octahedron. Only now the centers of the faces of the first will become the vertices for the second.
Method 1: Calculating the area of a cube based on its edge
In order to calculate the entire surface area of a cube, you will need to know one of its elements. The easiest way to solve it is when you know its edge or, in other words, the side of the square of which it consists. Usually this value is denoted by the Latin letter “a”.
Now you need to remember the formula that calculates the area of a square. To avoid confusion, its designation is introduced by the letter S 1.
For convenience, it is better to assign numbers to all formulas. This one will be the first.
But this is the area of only one square. There are six of them in total: 4 on the sides and 2 on the bottom and top. Then the surface area of the cube is calculated using the following formula: S = 6 * a 2. Her number is 2.
Method 2: how to calculate the area if the volume of the body is known
From the mathematical expression for the volume of a hexahedron, one can use it to calculate the length of the edge. Here she is:
The numbering continues, and here there is already the number 3.
Now it can be calculated and substituted into the second formula. If you follow the rules of mathematics, you need to derive the following expression:
This is a formula for the area of the entire surface of a cube, which can be used if the volume is known. This entry number is 4.
Method 3: Calculate the diagonal area of a cube
This is formula No. 5.
From it it is easy to derive an expression for the edge of a cube:
This is the sixth formula. After calculating it, you can again use the formula under the second number. But it’s better to write this:
It turns out to be numbered 7. If you look closely, you will notice that the last formula is more convenient than a step-by-step calculation.
Method 4: How to Use the Radius of an Inscribed or Circumscribed Circle to Calculate the Area of a Cube
If we denote the radius of the circle circumscribed around the hexahedron by the letter R, then the surface area of the cube will be easy to calculate using the following formula:
Its serial number is 8. It is easily obtained due to the fact that the diameter of the circle completely coincides with the main diagonal.
By denoting the radius of the inscribed circle with the Latin letter r, we can obtain the following formula for the area of the entire surface of the hexahedron:
This is formula No. 9.
A few words about the lateral surface of the hexahedron
If the problem requires finding the area of the lateral surface of a cube, then you need to use the technique already described above. When the edge of the body has already been given, then simply the area of the square needs to be multiplied by 4. This figure appeared due to the fact that the cube has only 4 side faces. The mathematical notation of this expression is as follows:
Its number is 10. If any other quantities are given, then proceed similarly to the methods described above.
Sample problems
Condition of the first. The surface area of the cube is known. It is equal to 200 cm². It is necessary to calculate the main diagonal of the cube.
1 way. You need to use the formula, which is indicated by the number 2. It will not be difficult to derive “a” from it. This mathematical notation will look like the square root of the quotient equal to S over 6. After substituting the numbers, we get:
a = √ (200/6) = √ (100/3) = 10 √3 (cm).
The fifth formula allows you to immediately calculate the main diagonal of the cube. To do this, you need to multiply the edge value by √3. It's simple. The answer turns out that the diagonal is 10 cm.
Method 2. In case you forgot the formula for the diagonal, but remember the Pythagorean theorem.
Similar to how it was in the first method, find the edge. Then you need to write the theorem for the hypotenuse twice: the first for the triangle on the face, the second for the one that contains the desired diagonal.
x² = a² + a², where x is the diagonal of the square.
d² = x² + a² = a² + a² + a² = 3 a². From this entry it is easy to see how the formula for the diagonal is obtained. And then all calculations will be the same as in the first method. It is a little longer, but allows you not to memorize the formula, but to get it yourself.
Answer: The diagonal of a cube is 10 cm.
Condition two. Using the known surface area, which is 54 cm2, calculate the volume of the cube.
Using the formula under the second number, you need to find out the value of the edge of the cube. How this is done is described in detail in the first method of solving the previous problem. Having carried out all the calculations, we find that a = 3 cm.
Now you need to use the formula for the volume of a cube, in which the length of the edge is raised to the third power. This means that the volume will be calculated as follows: V = 3 3 = 27 cm 3.
Answer: the volume of the cube is 27 cm3.
Condition three. You need to find the edge of the cube for which next condition. When an edge increases by 9 units, the area of the entire surface increases by 594.
Since no explicit numbers are given in the problem, only the difference between what was and what became, additional notation must be introduced. It is not difficult. Let the desired value be equal to “a”. Then the enlarged edge of the cube will be equal to (a + 9).
Knowing this, you need to write the formula for the surface area of a cube twice. The first one - for the initial value of the edge - will coincide with the one numbered 2. The second one will be slightly different. In it, instead of “a” you need to write the sum (a + 9). Since in the problem we're talking about about the difference in areas, then you need to subtract from larger area smaller:
6 * (a + 9) 2 - 6 * a 2 = 594.
Transformations need to be made. First, take the 6 on the left side of the equation out of brackets, and then simplify what remains in brackets. Namely (a + 9) 2 - a 2. The difference of squares is written here, which can be transformed as follows: (a + 9 - a)(a + 9 + a). After simplifying the expression, we get 9(2a + 9).
Now it needs to be multiplied by 6, that is, the number that was before the bracket, and equated to 594: 54(2a + 9) = 594. This is a linear equation with one unknown. It's easy to solve. First you need to open the brackets, and then move the term with an unknown value to the left side of the equality, and the numbers to the right. The resulting equation is: 2a = 2. From it it is clear that the desired value is equal to 1.
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 total area all sides on its surface. All six of its faces are proportional, therefore, knowing the length of one of them, you can calculate lateral area and the 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 same sizes. 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. Be sure 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, it is necessary to calculate in reverse order by dividing the value by 6 and then extracting 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. Just enter desired value parties 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.
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. full area cube surface and give specific examples.
Performed according to the formula SA = 6a 2. A cube (regular hexahedron) is one of 5 types of regular polyhedra, which is a regular rectangular parallelepiped, 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
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The 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. Be sure 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, basic course 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 edge length can be easily calculated. There's enough here basic knowledge. 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 is meant a regular polyhedron, all of whose faces are formed by regular quadrilaterals - 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?
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.
It is customary to measure volumes in cubic meters various substances V 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)
Helpful 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
