Knowing some parameters of the cube, you can easily detect its edge. To do this, it is enough just to have information about its volume, the area of a face or the length of the diagonal of a face or cube.
You will need
- Calculator
Instructions
1. There are mainly four types of problems in which you need to find the edge of a cube. This is the determination of the length of a cube edge by the area of the cube face, by the volume of the cube, by the diagonal of the cube face and by the diagonal of the cube. Let's look at all four options for such problems. (The rest of the tasks, as usual, are variations of the above or trigonometry problems that are extremely indirectly related to the issue under consideration) If the area of the face of the cube is known, then it is very easy to find the edge of the cube. Because the face of a cube is a square with a side equal to the edge of the cube, its area is equal to the square of the edge of the cube. Consequently, the length of a cube edge is equal to the square root of the area of its face, that is: a =?S, where a is the length of the cube edge, S is the area of the cube face.
2. Finding the face of a cube based on its volume is even easier. Considering that the volume of a cube is equal to the cube (third power) of the length of the edge of the cube, we find that the length of the edge of the cube is equal to the cube root (third power) of its volume, i.e.: a = ?V (cube root), where a is the length of the edge of the cube ,V – volume of the cube.
3. It is a little more difficult to find the length of the edge of a cube using the famous lengths of the diagonals. Let us denote by: a – the length of the edge of the cube; b – the length of the diagonal of the cube’s face; c – the length of the diagonal of the cube. As can be seen from the figure, the diagonal of the face and the edge of the cube form a right-angled equilateral triangle. Consequently, according to the Pythagorean theorem: a^2+a^2=b^2(^ is the sign of exponentiation). From there we find: a=?(b^2/2) (in order to find the edge of the cube, it is necessary to extract the square root of half square of the diagonal of the face).
4. In order to detect the edge of the cube along its diagonal, we will again use the drawing. The diagonal of the cube (c), the diagonal of the face (b) and the edge of the cube (a) form a right triangle. This means, according to the Pythagorean theorem: a^2+b^2=c^2. Let's use the above-established relationship between a and b and substitute b^2=a^2+a^2 into the formula. We get: a^2+a^2+a^2=c^2, from where we find: 3*a^2=c^2, therefore: a=?(c^2/3).
A cube is a rectangular parallelepiped with all edges equal. Consequently, the general formula for the volume of a rectangular parallelepiped and the formula for its surface area in the case Cuba are simplified. Also volume Cuba and him square surface can be detected by knowing the volume of a ball inscribed in it or a ball circumscribed around it.
You will need
- length of a side of a cube, radius of an inscribed and circumscribed sphere
Instructions
1. The volume of a rectangular parallelepiped is equal to: V = abc – where a, b, c are its dimensions. Consequently, the volume Cuba is equal to V = a*a*a = a^3, where a is the length of the side Cuba.Surface area Cuba equal to the sum of the areas of all its faces. Everyone has Cuba six faces, therefore square its surface is equal to S = 6*(a^2).
2. Let the ball be inscribed in a cube. Apparently, the diameter of this ball will be equal to the side Cuba. Substituting the length of the diameter into the expressions for volume instead of the length of the edge Cuba and applying that the diameter is equal to twice the radius, we then obtain 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. Surface area Cuba then it will be equal to S = 6*(d^2) = 24*(r^2).
3. Let the ball be described around Cuba. Then its diameter will coincide with the diagonal Cuba. Diagonal Cuba passes through the center Cuba and connects its two opposite points. First, look at one of the faces Cuba. The edges of this face are the legs of a right triangle, in which the diagonal of face d will be the hypotenuse. Then, according to the Pythagorean theorem, we get: d = sqrt((a^2)+(a^2)) = sqrt(2)*a.
4. After this, look at the triangle in which the hypotenuse will be the diagonal Cuba, and the diagonal of the face d and one of the edges Cuba a – its legs. Similarly, by the Pythagorean theorem we get: D = sqrt((d^2)+(a^2)) = sqrt(2*(a^2)+(a^2)) = a*sqrt(3). It turns out that according to the derived formula, diagonal Cuba is equal to D = a*sqrt(3). Hence, a = D/sqrt(3) = 2R/sqrt(3). Consequently, V = 8*(R^3)/(3*sqrt(3)), where R is the radius of the circumscribed sphere. Surface area Cuba is equal to S = 6*((D/sqrt(3))^2) = 6*(D^2)/3 = 2*(D^2) = 8*(R^2).
A cube is a three-dimensional polygon with six faces of a positive shape - a regular hexahedron. The number of positive faces determines the shape of each of them - these are squares. This is probably the most convenient of the polyhedral figures from the point of view of determining its geometric properties in the three-dimensional coordinate system we are familiar with. All its parameters can be calculated knowing only the length of one edge.
Instructions
1. If you have some physical object in the form Cuba, then to calculate its volume, measure the length of each face, and then use the algorithm described in the next step. If such a measurement is unrealistic, then you can, say, try to determine the volume of displaced water by placing a given cubic object in it. If you can find out the number of displaced water in liters, then the result can be converted into cubic decimeters - one liter in the SI system is equal to one cubic decimeter.
2. Raise the famous value of edge length to the third power Cuba, that is, the length of the side of the square that makes up each of its faces. Utilitarian calculations can be made on any calculator or with the help of the Google search engine. If you enter, say, “3.14 cubed” into the search query field, the search engine will immediately (without pressing a button) show the result.
3. If only the length of the diagonal is known Cuba, then this is also absolutely enough to calculate its volume. The diagonal of a positive octahedron is a segment connecting its two opposite vertices relative to the center. The length of such a diagonal can be expressed through the Pythagorean theorem as the length of the edge Cuba, divided by the root of 3. It follows that to find the volume Cuba you need to divide its diagonal by the root of 3 and build the result into a cube.
4. Similarly, you can calculate the volume Cuba, knowing only the length of the diagonal of its face. From the same Pythagorean theorem it follows that the length of the edge Cuba equal to the diagonal of the face divided by the root of 2. The volume in this case can be calculated by dividing the known length of the diagonal of the edge by the root of 2 and constructing the total in cube.
5. Do not forget about the dimension of the resulting result - if you calculate the volume based on the known dimensions in centimeters, then the result will be obtained in cubic centimeters. One decimeter contains ten centimeters, and one cubic decimeter (liter) contains 1000 (ten cubed) cubic centimeters. Accordingly, to convert the total to cubic decimeters, you need to divide the resulting value in centimeters by 1000.
Video on the topic
We continue to consider tasks with cubes and parallelepipeds. Basic formulas can be found at the beginning. The problems presented below are related to changes in volume and surface area as the rib increases (decreases).
One of the problems uses the concept of equal area. What does this mean? Equal-sized bodies are bodies that have equal volume. For example, if it is said that a ball is equal in size to a cube, this means that the ball and the cube have equal volume. Let's consider the tasks:
If each edge of a cube is increased by 9, its surface area will increase by 594. Find the edge of the cube.
Since there is a dependence of the surface area of a cube on its edge, then, of course, we will use the formula for the surface area of a cube:
It is said that when an edge increases by 9, the surface area increases by 594. Let us write the formula for the surface area for an enlarged cube:
The edge of a cube is equal to 1.
Answer: 1
Three edges of a rectangular parallelepiped coming from one vertex are equal to 4, 16, 27. Find an edge of a cube of equal size.
An equal-area cube is a cube whose volume is equal to the volume of a parallelepiped. It is known that the volume of a cube is found by the formula:
This means that if we find the volume of a parallelepiped, we can find the edge of a cube. The volume of the parallelepiped is equal to:
Thus:
*You can see how to extract the third root of a large number.
Answer: 12
How many times will the volume of a cube increase if its edges are increased sixfold?
Volume of a cube with an edge a equals V 1 = a 3 .
The volume of a cube with an edge six times larger is V 2 = (6a) 3 .
Let's divide V 2 on V 1 and we get the desired value:
The volume of the cube will increase 216 times.
Answer: 216
If each edge of a cube is increased by 3, then its volume will increase by 819. Find the edge of the cube.
Let the edge of the cube be equal a.
Let's write down what the volume is equal to for the original cube and for the enlarged one:
Volume of a cube with an edgea equals V 1 = a 3 .
Volume of a cube with an edge a+ 3 equals V 2 = (a + 3) 3 .
It is said that the volume increased by 819, which means:
Let's solve the equation:
Suitable value a= 8. A negative value for this problem has no physical meaning. Thus, the edge of the cube is 8.
Answer: 8
How many times will the surface area of a cube increase if its edge is increased by 24 times?
Let us write down the formula for the surface area of the original cube and the formula for the surface area for a cube with an enlarged edge:
Now all that remains is to find the area ratio:
Thus, the surface area will increase by 576 times.
Answer: 576
The volume of one cube is 729 times the volume of another cube. How many times is the surface area of the first cube greater than the surface area of the second cube?
Note that the first cube is a larger cube, the second is a smaller cube. We can easily solve this problem if we determine how many times the edge of the first cube is larger than the edge of the second. Let the edge of the small (second) cube be equal to x, and the edge of the larger cube to be y. Then
By condition:
Means
We found that the edge of the first cube is 9 times larger than the edge of the second, that is
Now let's write down the surface area for both cubes:
27080. Three edges of a rectangular parallelepiped emerging from one vertex are equal to 4, 6, 9. Find an edge of a cube of equal size.
27081. How many times will the volume of a cube increase if its edges are tripled?
27102. If each edge of a cube is increased by 1, then its volume will increase by 19. Find the edge of the cube.
27168. The volume of one cube is 8 times the volume of another cube. How many times is the surface area of the first cube greater than the surface area of the second cube?
There is also an excellent approach for solving problems in which we are talking about changing the volume and surface area for such bodies as: a cube, a parallelepiped, a ball, a regular quadrangular pyramid, a cone, a cylinder, while increasing (decreasing) the edge (radius) by a certain amount once. Such tasks can practically be solved in one line. I’ll tell you about this in the future, don’t miss it!
All the best! Good luck to you!
Sincerely, Alexander.
P.S: I would be grateful if you tell me about the site on social networks.
Knowing some parameters of a cube, you can easily find its edge. To do this, it is enough just to have information about its volume, the area of a face or the length of the diagonal of a face or cube.
You will need
- Calculator
Instructions
There are mainly four types of problems in which you need to find the edge of a cube. This is the determination of the length of a cube edge by the area of the cube face, by the volume of the cube, by the diagonal of the cube face and by the diagonal of the cube. Let's consider all four variants of such problems. (The remaining tasks, as a rule, are variations of the above or trigonometry tasks that are very indirectly related to the issue at hand)
If the area of a cube face is known, then finding the edge of the cube is very simple. Since the face of a cube is a square with a side equal to the edge of the cube, its area is equal to the square of the edge of the cube. Therefore, the length of the edge of a cube is equal to the square root of the area of its face, that is:
a is the length of the cube edge,
S is the area of the cube face.
Finding the face of a cube based on its volume is even easier. Considering that the volume of a cube is equal to the cube (third power) of the length of the edge of the cube, we find that the length of the edge of the cube is equal to the cube root (third power) of its volume, i.e.:
a=?V (cube root), where
a is the length of the cube edge,
V is the volume of the cube.
It is a little more difficult to find the length of the edge of a cube using the known lengths of the diagonals. Let's denote by:
a is the length of the edge of the cube-
b - length of the diagonal of the cube face -
c is the length of the diagonal of the cube.
As can be seen from the figure, the diagonal of the face and the edges of the cube form a right-angled equilateral triangle. Therefore, according to the Pythagorean theorem:
(^ is the symbol for exponentiation).
From here we find:
(to find the edge of a cube you need to take the square root of half the square of the diagonal of the face).
To find the edge of the cube along its diagonal, we will again use the figure. The diagonal of the cube (c), the diagonal of the face (b) and the edge of the cube (a) form a right triangle. So, according to the Pythagorean theorem:
Let's use the above-established relationship between a and b and substitute it into the formula
b^2=a^2+a^2. We get:
a^2+a^2+a^2=c^2, from where we find:
3*a^2=c^2, therefore.
Lengths of a cube edge by the area of the cube face, by the volume of the cube, by the diagonal of the cube face and by the diagonal of the cube. Let's consider all four variants of such problems. (The rest of the tasks, like , are variations of the above or trigonometry tasks that are very indirectly related to the issue at hand)
If the area of a cube face is known, then finding the edge of the cube is very simple. Since the face of a cube is a square with a side equal to the edge of the cube, its area is equal to the square of the edge of the cube. Therefore, the length of the edge of a cube is equal to the square root of the area of its face, that is:
a is the length of the cube edge,
S is the area of the cube face.
Finding the face of a cube based on its volume is even easier. Considering that the volume of a cube is the cube (of the third power) of the length of the edge of the cube, we find that the length of the edge of the cube is equal to the cubic root (of the third power) of its volume, i.e.:
a is the length of the cube edge,
V is the volume of the cube.
It is a little more difficult to find the length of the edge of a cube using the known lengths of the diagonals. Let's denote by:
a is the length of the cube edge;
b is the length of the diagonal of the cube face;
c is the length of the diagonal of the cube.
As can be seen from the figure, the diagonal of the face and the edges of the cube form a rectangular shape. Therefore, according to the Pythagorean theorem:
From here we find:
(to find the edge of a cube you need to take the square root of half the square of the diagonal of the face).
To find the edge of the cube along its diagonal, we will again use the figure. The diagonal of the cube (c), the diagonal of the face (b) and the edge of the cube (a) form a right triangle. So, according to the Pythagorean theorem:
Let's use the above-established relationship between a and b and substitute it into the formula
b^2=a^2+a^2. We get:
a^2+a^2+a^2=c^2, from where we find:
3*a^2=c^2, therefore:
Sources:
- cube edge drawing
A cube is a rectangular parallelepiped with all edges equal. Therefore, the general formula for the volume of a rectangular parallelepiped and the formula for its surface area in the case Cuba are simplified. Also volume Cuba and him square surfaces can be found by knowing the volume of a sphere inscribed in it or a sphere circumscribed around it.
You will need
- length of a side of a cube, radius of an inscribed and circumscribed sphere
Instructions
The volume is equal to: V = abc - where a, b, c are its . That's why Cuba is equal to V = a*a*a = a^3, where a is the length of the side Cuba.Surface area Cuba equal to the sum of the areas of all its faces. Total Cuba six faces, so square its surface is equal to S = 6*(a^2).
Let the ball be inscribed in a cube. Obviously, the diameter of this ball will be equal to the side Cuba. Substituting length in expressions for instead of edge length Cuba and using that the diameter is equal to twice, we then obtain 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. Surface area Cuba then it will be equal to S = 6*(d^2) = 24*(r^2).
Let the ball be described Cuba. Then its diameter will coincide with the diagonal Cuba. Diagonal Cuba passes through the center Cuba and connects its two points.
First, consider one of the faces Cuba. The edges of this face are legs, in which the diagonal of face d will be the hypotenuse. Then, by the Pythagorean theorem, we get: d = sqrt((a^2)+(a^2)) = sqrt(2)*a.
Then consider a triangle in which the hypotenuse is the diagonal Cuba, and the diagonal of the face d and one of the edges Cuba a - its legs. Similarly, by the Pythagorean theorem we get: D = sqrt((d^2)+(a^2)) = sqrt(2*(a^2)+(a^2)) = a*sqrt(3).
So, according to the derived formula, the diagonal Cuba 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 circumscribed sphere. Surface area Cuba is equal to S = 6*((D/sqrt(3))^2) = 6*(D^2)/3 = 2*(D^2) = 8*(R^2).
Sources:
- the volume of the cube is
A cube is a three-dimensional polygon with six faces of regular shape - a regular hexahedron. The number of regular faces determines the shape of each of them - these are squares. This is perhaps the most convenient of the polyhedral figures from the point of view of determining its geometric properties in the three-dimensional coordinate system familiar to us. All its parameters can be calculated knowing just the length of one edge.
Instructions
If you have some physical object in the form Cuba, then to calculate its volume, measure the length of any face, and then use the algorithm described in the next step. If measurement is impossible, then you can, for example, try to determine the volume of displaced water by placing this cubic object in it. If you can find out the amount of displaced water in liters, then the result can be converted into cubic s - one liter in the SI system is equal to one cubic decimeter.
Raise the known edge length to the third power Cuba, that is, the length of the side
The problems presented below are simple, most of them can be solved in 1 step. In this article we will consider a rectangular parallelepiped (all faces are rectangles). What do you need to know and understand? First, look at the formulas for the volume and surface area of a cube and a rectangular parallelepiped, as well as the diagonal formula, you can.Let us briefly list the formulas:
Rectangular parallelepiped
Let the edges be equal A,b, With.
Surface area:
Volume:
Diagonal:
Cube
Let the edge of the cube be equal A.
Surface area:
Volume:
Diagonal:
*It is clear that the formulas of a cube are a consequence of the corresponding formulas of a rectangular parallelepiped. A cube is a parallelepiped in which all edges are equal and the faces are squares.
Let's consider the tasks:
The two edges of a cuboid coming from the same vertex are 5 and 8. The surface area of this cuboid is 210. Find the third edge coming from the same vertex.
Let us denote the known edges as A And b, and the unknown for c.
Then the formula for the surface area of a parallelepiped is expressed as:
All that remains is to substitute the data and solve the equation:
Answer: 5
The surface area of a cube is 200. Find its diagonal.
Let's construct the diagonal of the cube:
The surface area of a cube is expressed in terms of its edge A How S = 6A 2, which means we can find the edge A:
The diagonal of the face of a cube according to the Pythagorean theorem is equal to:
The diagonal of a cube according to the Pythagorean theorem is equal to:
Then
*You could immediately use the cube diagonal formula:
Answer: 10
The volume of the cube is 343. Find its surface area.
The surface area of a cube is expressed in terms of its edgeA How S = 6 A 2 and the volume is V = A 3 . So we can find the edge of the cube and then calculate the surface area:
Thus, the surface area of the cube is:
Answer: 294
27060. The two edges of a cuboid extending from the same vertex are 1 and 2. The surface area of the cuboid is 16. Find its diagonal.
The diagonal of a parallelepiped is calculated by the formula:
where a, b and c are edges.
Let's find the third edge. We can do this using the formula for the surface area of a parallelepiped:
We substitute the data and solve the equation:
Thus, the diagonal will be equal to:
Answer: 3
27063. Find the lateral edge of a regular quadrangular prism if the side of its base is 20 and its surface area is 1760.
At the base of a regular quadrangular prism is a square. It is clear that it is a parallelepiped. The same formulas apply. Let the side edge be equal to x. We can find it using the surface area formula:
Answer: 12
A regular quadrangular prism with a base side of 0.8 and a side edge of 1 is cut from a unit cube. Find the surface area of the remaining part of the cube.
A unit cube is a cube with edge equal to 1.
The surface area of the resulting polyhedron can be calculated as follows: from the surface area of the cube, you need to subtract two areas of the base of the cut out prism and add four areas of the side face of the cut out prism with sides 1 and 0.8:
Answer: 7.92
The area of the face of a rectangular parallelepiped is 48. The edge perpendicular to this face is 8. Find the volume of the parallelepiped.
It is enough to apply the volume formula........................
The volume of a rectangular parallelepiped is equal to the product of its three edges, or the product of the area of the base and the height. In this case, the role of the base is played by the edge, the role of the height is played by the edge, which is perpendicular to it. We get:
Answer: 384
You will solve the following problems without difficulty.
27077. The volume of a rectangular parallelepiped is 64. One of its edges is 4. Find the area of the face of the parallelepiped perpendicular to this edge. Answer: 16.
27078. The volume of a rectangular parallelepiped is 60. The area of one of its faces is 12. Find the edge of the parallelepiped perpendicular to this face. Answer: 5.
27079. Two edges of a rectangular parallelepiped emerging from the same vertex are 8 and 6. The volume of the parallelepiped is 240. Find the third edge of the parallelepiped emerging from the same vertex. Answer: 4.
More for your own solution:
27054. The two edges of a cuboid coming from the same vertex are 3 and 4. The surface area of this cuboid is 94. Find the third edge coming from the same vertex.
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