Students often ask indignantly: “How will this be useful to me in life?” On any topic of each subject. The topic about the volume of a parallelepiped is no exception. And this is where you can just say: “It will come in handy.”
How, for example, can you find out whether a package will fit in a postal box? Of course, you can choose the right one by trial and error. What if this is not possible? Then calculations will come to the rescue. Knowing the capacity of the box, you can calculate the volume of the parcel (at least approximately) and answer the question posed.
Parallelepiped and its types
If we literally translate its name from ancient Greek, it turns out that this is a figure consisting of parallel planes. There are the following equivalent definitions of a parallelepiped:
- a prism with a base in the form of a parallelogram;
- a polyhedron, each face of which is a parallelogram.
Its types are distinguished depending on what figure lies at its base and how the lateral ribs are directed. IN general case talk about inclined parallelepiped, whose base and all faces are parallelograms. If the previous type side faces become rectangles, then it will need to be called direct. And rectangular and the base also has 90º angles.
Moreover, in geometry they try to depict the latter in such a way that it is noticeable that all the edges are parallel. Here, by the way, is the main difference between mathematicians and artists. It is important for the latter to convey the body in compliance with the law of perspective. And in this case, the parallelism of the ribs is completely invisible.
About the introduced notations
In the formulas below, the notations indicated in the table are valid.
Formulas for an inclined parallelepiped
First and second for areas:
The third is to calculate the volume of a parallelepiped:
Since the base is a parallelogram, to calculate its area you will need to use the appropriate expressions.
Formulas for a rectangular parallelepiped
Similar to the first point - two formulas for areas:
And one more for volume:
First task
Condition. Given a rectangular parallelepiped, the volume of which needs to be found. The diagonal is known - 18 cm - and the fact that it forms angles of 30 and 45 degrees with the plane of the side face and the side edge, respectively.
Solution. To answer the problem question, you will need to know all the sides in three right triangles. They will give required values edges along which you need to calculate the volume.
First you need to figure out where the 30º angle is. To do this, you need to draw a diagonal of the side face from the same vertex from where the main diagonal of the parallelogram was drawn. The angle between them will be what is needed.
The first triangle that will give one of the values of the sides of the base will be the following. It contains the required side and two drawn diagonals. It's rectangular. Now we need to use the relation opposite leg(base sides) and hypotenuse (diagonals). It is equal to the sine of 30º. That is, the unknown side of the base will be determined as the diagonal multiplied by the sine of 30º or ½. Let it be designated by the letter “a”.
The second will be a triangle containing a known diagonal and an edge with which it forms 45º. It is also rectangular, and you can again use the ratio of the leg to the hypotenuse. In other words, side edge to diagonal. It is equal to the cosine of 45º. That is, “c” is calculated as the product of the diagonal and the cosine of 45º.
c = 18 * 1/√2 = 9 √2 (cm).
In the same triangle you need to find another leg. This is necessary in order to then calculate the third unknown - “in”. Let it be designated by the letter “x”. It can be easily calculated using the Pythagorean theorem:
x = √(18 2 - (9√2) 2) = 9√2 (cm).
Now we need to consider another right triangle. It already contains known parties“c”, “x” and the one that needs to be counted, “b”:
in = √((9√2) 2 - 9 2 = 9 (cm).
All three quantities are known. You can use the formula for volume and calculate it:
V = 9 * 9 * 9√2 = 729√2 (cm 3).
Answer: the volume of the parallelepiped is 729√2 cm 3.
Second task
Condition. You need to find the volume of a parallelepiped. In it, the sides of the parallelogram, which lies at the base, are known to be 3 and 6 cm, as well as its acute angle - 45º. The side rib has an inclination to the base of 30º and is equal to 4 cm.
Solution. To answer the question of the problem, you need to take the formula that was written for the volume inclined parallelepiped. But both quantities are unknown in it.
The area of the base, that is, of a parallelogram, will be determined by a formula in which you need to multiply the known sides and the sine of the acute angle between them.
S o = 3 * 6 sin 45º = 18 * (√2)/2 = 9 √2 (cm 2).
The second unknown quantity is height. It can be drawn from any of the four vertices above the base. It can be found from a right triangle in which the height is the leg and the side edge is the hypotenuse. In this case, an angle of 30º lies opposite unknown height. This means that we can use the ratio of the leg to the hypotenuse.
n = 4 * sin 30º = 4 * 1/2 = 2.
Now all the values are known and the volume can be calculated:
V = 9 √2 * 2 = 18 √2 (cm 3).
Answer: the volume is 18 √2 cm 3.
Third task
Condition. Find the volume of a parallelepiped if it is known that it is straight. The sides of its base form a parallelogram and are equal to 2 and 3 cm. Acute angle there is 60º between them. The minor diagonal of a parallelepiped is larger diagonal grounds.
Solution. In order to find out the volume of a parallelepiped, we use the formula with the base area and height. Both quantities are unknown, but they are easy to calculate. The first one is height.
Since the smaller diagonal of the parallelepiped is the same size as larger base, then they can be designated by one letter d. Larger angle a parallelogram is 120º, since it forms 180º with an acute one. Let the second diagonal of the base be designated by the letter “x”. Now for the two diagonals of the base we can write the cosine theorems:
d 2 = a 2 + b 2 - 2av cos 120º,
x 2 = a 2 + b 2 - 2ab cos 60º.
It makes no sense to find values without squares, since later they will be raised to the second power again. After substituting the data, we get:
d 2 = 2 2 + 3 2 - 2 * 2 * 3 cos 120º = 4 + 9 + 12 * ½ = 19,
x 2 = a 2 + b 2 - 2ab cos 60º = 4 + 9 - 12 * ½ = 7.
Now the height, which is also the side edge of the parallelepiped, will turn out to be a leg in the triangle. The hypotenuse will be known diagonal body, and the second leg - “x”. We can write the Pythagorean Theorem:
n 2 = d 2 - x 2 = 19 - 7 = 12.
Hence: n = √12 = 2√3 (cm).
Now the second unknown quantity is the area of the base. It can be calculated using the formula mentioned in the second problem.
S o = 2 * 3 sin 60º = 6 * √3/2 = 3√3 (cm 2).
Combining everything into the volume formula, we get:
V = 3√3 * 2√3 = 18 (cm 3).
Answer: V = 18 cm 3.
Fourth task
Condition. It is required to find out the volume of a parallelepiped that meets the following conditions: the base is a square with a side of 5 cm; the side faces are rhombuses; one of the vertices located above the base is equidistant from all the vertices lying at the base.
Solution. First you need to deal with the condition. There are no questions with the first point about the square. The second, about rhombuses, makes it clear that the parallelepiped is inclined. Moreover, all its edges are equal to 5 cm, since the sides of the rhombus are the same. And from the third it becomes clear that the three diagonals drawn from it are equal. These are two that lie on the side faces, and the last one is inside the parallelepiped. And these diagonals are equal to the edge, that is, they also have a length of 5 cm.
To determine the volume, you will need a formula written for an inclined parallelepiped. There are again no known quantities in it. However, the area of the base is easy to calculate because it is a square.
S o = 5 2 = 25 (cm 2).
The situation with height is a little more complicated. It will be like this in three figures: a parallelepiped, quadrangular pyramid And isosceles triangle. This last circumstance should be taken advantage of.
Since it is the height, it is a leg in right triangle. The hypotenuse in it will be famous rib, and the second leg equal to half diagonals of the square (height is also the median). And the diagonal of the base is easy to find:
d = √(2 * 5 2) = 5√2 (cm).
The height will need to be calculated as the difference between the second power of the edge and the square of half the diagonal and then remember to take the square root:
n = √ (5 2 - (5/2 * √2) 2) = √(25 - 25/2) = √(25/2) = 2.5 √2 (cm).
V = 25 * 2.5 √2 = 62.5 √2 (cm 3).
Answer: 62.5 √2 (cm 3).
A parallelepiped is a prismatic figure, all of whose faces are parallelograms. If ordinary rectangles act as faces, then the parallelepiped is rectangular and it is the shape of this figure that such real objects like panel houses, aquariums, books, printers or bricks.
Parallelepiped geometry
A rectangular parallelepiped is limited by six faces, while opposite faces the figures are equal and parallel to each other. This geometric figure represents special case direct quadrangular prism. The parallelepiped has 12 edges and 8 vertices. At each of the vertices, three edges of the figure converge, which are the length, width and height of the parallelepiped or its dimensions. If the length, width and height of the figure are equal, then the parallelepiped turns into a cube.
Parallelepipeds in real life
A large number of objects that exist in reality have the shape of a parallelepiped. This form has become widespread due to ease of production, ease of storage and transportation, ideal compatibility of identical parallelepipeds, stability and consistency of size. Objects such as bricks, boxes, smartphones, power supplies, houses, rooms and much more have a parallelepiped shape.
Volume of a parallelepiped
An important property of any geometric body is its capacity, that is, the volume of the figure. Volume is a characteristic of an object that shows how many unit cubes it can hold. In general, the volume of any prismatic figure is calculated by the formula:
where So is the area of the base of the figure, and h is its height.
This formula is easily illustrated following example. Imagine that you have one sheet of A4 paper. This is an ordinary rectangle, which is characterized strictly a certain area. Roughly speaking, a sheet is a plane. Now imagine a standard pack of paper of 500 A4 sheets. It's already volumetric figure, having the shape of a parallelepiped. It’s easy to find out its volume; just multiply the area of the sheet lying at the base by their number, that is, by the height of the prism.
A parallelepiped is a special case of a prism, the base of which is a rectangle. The area of a rectangle is the simple product of its sides, therefore for a parallelepiped:
To determine the volume, just multiply So by the height of the figure. Thus, the volume rectangular parallelepiped is calculated using a simple formula representing the multiplication of three sides of a body:
V = a × b × h,
where a is the length, b is the width, h is the height of the geometric figure.
To determine the volume of a rectangular parallelepiped, you just need to measure these three parameters and simply multiply them. If you don’t want to constantly keep in your head formulas for determining the volumes and areas of geometric shapes, then use our catalog of online calculators: each tool will tell you what parameters you should measure and instantly calculate the result. Let's look at a couple of examples when you may need to determine the volume of a parallelepiped.
Examples from life
Aquarium
For example, you bought an old aquarium in the shape of a parallelepiped, but no one told you how much volume this structure has. Aquarium volume - important parameter, which determines the power of the heating system for sea creatures. Calculate this characteristic It’s not difficult - just measure the length, width and height of the aquarium and enter this data into the calculator form. Let's say the length of the aquarium is 1 m, the width is 50 cm, and the height is 70 cm. For correct calculation, it is important to express all sides in the same units of measurement, for example, in meters.
V = 1 × 0.5 × 0.7 = 0.35
Thus, the volume of the aquarium will be 0.35 cubic meters or 350 liters. Knowing the volume, you can easily select the power for the heating system.
Construction
Let's say you are pouring a slab foundation for your dacha and you need to find out how much concrete will be needed to pour the foundation. A slab foundation is a solid monolithic slab that is located under the entire area of the building. In order to find out the required volume of concrete, it is necessary to calculate the volume of the slab. The slab, fortunately, has the shape of a rectangular parallelepiped, so you can easily calculate the required amount of concrete. Let's say your dacha is a standard house 6 by 6 meters. You already know two of the three required parameters. According to the requirements, the thickness of the slab foundation must be at least 10 cm, and you can choose the appropriate size yourself. For example, you decide to pour a slab 20 cm thick. For correct calculation, set all parameters in the same units of measurement, that is, meters, and get the result:
V = 6 × 6 × 0.2 = 7.2
Therefore, to pour the foundation you will need 7.2 cubic meters of concrete.
Conclusion
Determining the volume of parallelepiped figures can be useful to you in many cases: from everyday problems to production issues, from school assignments to design tasks. Our online calculator will help you solve problems of any complexity.
>> Lesson 31. Formula for the volume of a rectangular parallelepiped
A rectangular parallelepiped is a spatial figure limited rectangles.
Many objects from the environment have a parallelepiped shape: a box, cubes, TV, wardrobe, etc..
Math lesson in 5th grade. (Vilenkin)
Subject: Volumes. Volume of a rectangular parallelepiped.
Target: 1. Consolidate knowledge on this topic when solving problems. Prepare for test work. Give the ratio of volume units.
2. Repeat the properties of multiplication, simplification of expressions, parts of a parallelepiped.
3. Educate environmental aspect, attention.
Equipment: on the board: topic, task for oral counting; handout: models of parallelepiped, cube, matchbox; for children: cheat sheets, rulers, two-color signal circles,
Progress of the lesson.
Organizational moment.
Good afternoon, happy hour, we have mathematics. On the desk: rulers, cheat sheets, notebooks, textbooks.
Oral counting (warm-up) No. 806 – in rows “in a chain”,
- apply distributive property multiplication:
(x + 8) 20 on the board
247 123 – 147 123
- simplify:
20a – 19a 4x + x – 2x
13v - 27 + 13v - 10v
Communicate the topic and purpose.
— What geometric figures did you get acquainted with? Today we will repeat how to find the volume of a rectangular parallelepiped and the units of volume. Getting ready for the test.
IV. Repetition of what has been learned. cube models,
— Show top, back, bottom and front edges. parallelepiped
— Show two faces that have a common edge,
— Show vertical edges.
(show 2 or 3 students at the same time)
Game "Yes - no"
— Any cube is a rectangular parallelepiped (+) signal
— A rectangular parallelepiped has 10 vertices (-, 8) circles
– 6 edges (+) – 12 edges (+)
— Each face of the cube is a square (+)
— If the length of a rectangular parallelepiped is not equal to its height, then it cannot be a cube (+)
— Volume of a rectangular parallelepiped equal to the product its three dimensions (+)
Find the formula.
- calculate the volume matchbox, cube, parallelepiped. visibility
— additional material“How much air does a person need to breathe?”
With each inhalation, a person introduces 9 liters of air into his lungs in 1 minute. This amounts to 9 * 60 per hour, i.e. 540 liters. Let’s round up to 500 liters or half a cubic meter and find out that a person inhales 12 m³ of air per day. This volume is 14 kg.
In one day, a person passes through his body more air than food: no one eats even 3 kg per day, but we inhale 14 kg. If we consider that the inhaled air consists of 4/5 nitrogen, which is useless for breathing, then it seems that our body consumes only 3 kg, i.e. approximately the same amount as food (solid and liquid).
Do you need any other proof of the need to renew the air in the living room?
- No. 804, 801 - on the board,
— How to calculate the volume of a parallelepiped or cube?
— In what units is volume measured?
VI. Ratio of volume units.“cheat sheets” Write in “cheat sheets”. flyleaf
— Game “The Weakest Link” — No. 802,
— Task on cards.
— Express in cubic cm:
6 dm³, 287 dm³
5 dm³ 23 cm³ 16000 mm³
5 dm³ 635 cm³ 2 dm³ 80 cm³
— Express in cubic dm:
6m³ 580cm³ 7m³ 15dm³
VII. Repetition of what has been learned. № 808
VIII. Result:— What do you remember from the lesson?
— Who worked for 5? by 4?
IX. Homework: § 21, No. 822 (a, b), No. 823.
Mathematics
5th grade
21. Volumes.
If you fill the mold with wet sand, and then turn it over and remove it, you will get figures that have the same volume (Fig. 83). If the mold is filled with water, the volume of water will be equal to volume each sand figure.
Rice. 83
To compare the volumes of two vessels, you can fill one of them with water and pour it into the second vessel. If the second vessel is filled and there is no water left in the first vessel, then the volumes of the vessels are equal. If water remains in the first vessel, then its volume is greater than the volume of the second vessel. And if it is not possible to fill the second vessel with water, then the volume of the first vessel is less than the volume of the second.
The following units are used to measure volumes: cubic millimeter (mm3), cubic centimeter (cm3), cubic decimeter (dm3), cubic meter (m3), cubic kilometer (km3).
For example: a cubic centimeter is the volume of a cube with an edge of 1 cm (Fig. 84).
Rice. 84
A cubic decimeter is also called a liter.
The figure in Figure 85 consists of 4 cubes with an edge of 1 cm. This means that its volume is 4 cm3.
Rice. 85
Let us derive a rule for calculating the volume of a rectangular parallelepiped.
Formulas for the volumes of parallelepipeds and cubes
Let a rectangular parallelepiped have a length of 4 cm, a width of 3 cm and a height of 2 cm (Fig. 86, a). Let's divide it into two layers 1 cm thick (Fig. 86, b). Each of these layers consists of 3 columns 4 cm long (Fig. 86, c), and each column consists of 4 cubes with an edge of 1 cm (Fig. 86, d). This means that the volume of each column is 4 cm3, each layer is 4 3 (cm3), and the entire rectangular parallelepiped is (4 3) 2, that is, 24 cm3.
Rice. 86
To find the volume of a rectangular parallelepiped, you need to multiply its length by its width and height.
The formula for the volume of a rectangular parallelepiped is
where V is volume; a, b, c - measurements.
If the edge of a cube is 4 cm, then the volume of the cube is 4 4 4 = 43 (cm3), that is, 64 cm3.
If the edge of a cube is equal to a, then the volume V of the cube is equal to a a a = a3.
This means that the formula for the volume of a cube has the form
That is why the entry a3 is called the cube of a.
The volume of a cube with an edge of 1 m is equal to 1 m3. And since 1 m = 10 dm, then 1 m3 = 103 dm3, that is, 1 m3 = 1000 dm3 = 1000 l.
In the same way we find that
1 l = 1 dm3 = 1000 cm3; 1 cm3 = 1000 mm3;
1 km3 = 1,000,000,000 m3 (see figure).
Self-test questions
- The figure consists of 19 cubes with a side of 1 cm each; what is the volume of this figure?
- What is a cubic centimeter; cubic meter?
- What is another name for cubic decimeter?
- How many cubic centimeters is 1 liter?
- How many liters is a cubic meter equal?
- How many cubic meters are in a cubic kilometer?
- Write the formula for the volume of a rectangular parallelepiped.
- What does the letter V mean in this formula; letters a, b, c?
- Write the formula for the volume of a cube.
Do the exercises
819. Figures are made from cubes with an edge of 1 cm (Fig. 87). Find the volumes and surface areas of these figures.
Rice. 87
820. Find the volume of a rectangular parallelepiped if:
- a) a = 6 cm, b = 10 cm, c = 5 cm;
- b) a = 30 dm, b = 20 dm, c = 30 dm;
- c) a = 8 dm, b = 6 m, c = 12 m;
- d) a = 2 dm 1 cm, b = 1 dm 7 cm, c = 8 cm;
- e) a = 3 m, b = 2 dm, c = 15 cm.
821. Square bottom edge of a rectangular parallelepiped is 24 cm2. Determine the height of this parallelepiped if its volume is 96 cm3.
822. The volume of the room is 60 m3. The height of the room is 3 m, the width is 4 m. Find the length of the room and the area of the floor, ceiling, and walls.
823. Find the volume of a cube whose edge is 8 dm; 3 dm 6 cm.
824. Find the volume of a cube if its surface area is 96 cm2.
825. Express:
- a) in cubic centimeters: 5 dm3 635 cm3; 2 dm3 80 cm3;
- b) in cubic decimeters: 6 m3 580 dm3; 7 m3 15 dm3;
- c) in cubic meters and decimeters: 3270 dm3; 12,540,000 cm3.
826. The height of the room is 3 m, width 5 m and length 6 m. How many cubic meters of air are in the room?
827. The length of the aquarium is 80 cm, the width is 45 cm, and the height is 55 cm. How many liters of water must be poured into this aquarium so that the water level is 10 cm below the top edge of the aquarium?
828. The rectangular parallelepiped (Fig. 88) is divided into two parts. Find the volume and surface area of the entire parallelepiped and both of its parts. Is the volume of a parallelepiped equal to the sum of the volumes of its parts? Can this be said about their surface areas? Explain why.
Rice. 88
829. Calculate orally:
830. Restore the chain of calculations:
831. Find the meaning of the expression:
- a) 23 + Z2;
- b) 33 + 52;
- c) 43 + 6;
- d) 103 - 10.
832. How many tens are there in the quotient:
- a) 1652: 7;
- b) 774: 6;
- c) 1632: 12;
- d) 2105: 5?
833. Do you agree with the statement:
- a) any cube is also a rectangular parallelepiped;
- b) if the length of a rectangular parallelepiped is not equal to its height, then it cannot be a cube;
- c) each face of a cube is a square?
834. Four identical barrels hold 26 buckets of water. How many buckets of water can 10 of these barrels hold?
835. In how many ways from 7 beads different colors can you make a necklace (with a clasp)?
836. Name in a rectangular parallelepiped (Fig. 89):
- a) two faces having a common edge;
- b) top, back, front and bottom edges;
- c) vertical ribs.
Rice. 89
837. Solve the problem:
- Find the area of each plot if the area of the first plot is 5 times more area the second, and the area of the second is 252 hectares less area first.
- Find the area of each plot if the area of the second plot is 324 hectares greater than the area of the first plot, and the area of the first plot is 7 times less than the area of the second.
838. Follow these steps:
- 668 (3076 + 5081);
- 783 (66 161 — 65 752);
- 2 111 022: (5960 — 5646);
- 2 045 639: (6700 — 6279).
839. In Rus', in the old days, a bucket (about 12 l), a shtof (a tenth of a bucket) was used as units of volume measurement; in the USA, England and other countries a barrel (about 159 l), a gallon (about 4 l), a bushel (about 36) were used. l), pint (from 470 to 568 cubic centimeters). Compare these units. Which ones are larger than 1 m3?
840. Find the volumes of the figures shown in Figure 90. The volume of each cube is 1 cm3.
Rice. 90
841. Find the volume of a rectangular parallelepiped (Fig. 91).
Rice. 91
842. Find the volume of a rectangular parallelepiped if its dimensions are 48 dm, 16 dm and 12 dm.
843. The barn, shaped like a rectangular parallelepiped, is filled with hay. The length of the barn is 10 m, width 6 m, height 4 m. Find the mass of hay in the barn if the mass of 10 m3 of hay is 6 quintals.
844. Express in cubic decimeters:
- 2 m3 350 dm3;
- 3 m3 7 dm3;
- 4 m3 30 dm3;
- 18,000 cm3;
- 210,000 cm3.
845. The volume of a rectangular parallelepiped is 1248 cm3. Its length is 13 cm and its width is 8 cm. Find the height of this parallelepiped.
846. Using the formula V = abc calculate:
- a) V, if a - 3 dm, b = 4 dm, c = 5 dm;
- b) a, if V = 2184 cm3, b = 12 cm, c = 13 cm;
- c) b, if V = 9200 cm3, a = 23 cm, c = 25 cm;
- d) ab, if V = 1088 dm3, c = 17 cm.
What is the meaning of ab?
847. Father older than my son for 21 years. Write down a formula expressing - the age of the father - through b - the age of the son. Find using this formula:
- a) a, if b = 10;
- b) a, if b = 18;
- c) b, if a = 48.
848. Find the meaning of the expression:
- a) 700,700 - 6054 (47,923 - 47,884) - 65,548;
- b) 66,509 + 141,400: (39,839 - 39,739) + 1985;
- c) (851 + 2331) : 74 - 34;
- d) (14,084: 28 - 23) 27 - 12,060;
- e) (102 + 112 + 122) : 73 + 895;
- f) 2555: (132 + 142) + 35.
849. Calculate from the table (Fig. 92):
- a) how many times does the number 9 appear;
- b) how many times do the numbers 6 and 7 appear in the table (not counting them separately);
- c) how many times do the numbers 5, 6 and 8 appear (not counting them individually).
Rice. 92
Stories about the history of the emergence and development of mathematics
200 years ago in different countries, including in Russia, were used various systems units for measuring length, mass and other quantities. The relationships between the measures were complex, there were different definitions for units of measurement.
For example, to this day in Great Britain there are two different “tons” (2000 and 2940 pounds), more than 50 different “bushels”, etc. This hampered the development of science and trade between countries, so there was a need to introduce a unified system of measures , convenient for all countries, with simple relationships between units.
Such a system - it was called the metric system of measures - was developed in France. Basic unit of length, 1 meter (from Greek word“metron” - measure), defined as a forty-millionth fraction of the Earth’s circumference, the basic unit of mass, 1 kilogram - as the mass of 1 dm3 clean water. The remaining units were determined through these two, the ratios between units of the same value were equal to 10, 100, 1000, etc.
The metric system of measures has been adopted by most countries of the world; in Russia its introduction began in 1899. Great contribution to the introduction and dissemination metric system measures in our country belong to Dmitry Ivanovich Mendeleev, the great Russian chemist.
However, according to tradition, even today the old units are sometimes used. sailors measure distances in miles (1852 m) and cables (a tenth of a mile, that is, about 185 m), speed - in knots (1 mph). The mass of diamonds is measured in carats (200 mg, that is, a fifth of a gram is the mass of a wheat grain). The volume of oil is measured in barrels (159 l), etc.
This can be done in different ways, it all depends on what quantities and objects we have.
So, the first method, which is suitable exclusively for a rectangular parallelepiped.
To determine the volume of a parallelepiped you will need its height, width and length.
Since rectangles form a parallelepiped, let's mark their length and width with the letters a and b, respectively. Then the area of the rectangle will be calculated as a*b.
The height of a parallelepiped is the height of the side edge, and since the height is a constant value, to find the volume you need to multiply the base area of the parallelepiped by the height. This is expressed by the following formula: V = a*b*c = S*c, where c is the height.
Let's look at an example. Let's say we have a parallelepiped with a base length and width of 5 and 8 cm, and its height is 11 cm. It is necessary to calculate the volume.
Find the area of the base: 5*8=40 sq. cm. Now we multiply the resulting value by the height 40*11=440 cubic meters. cm is the volume of the figure.
Second way.
Since the base of a parallelepiped is the geometric figure of a parallelogram, you need to determine its area. To find the area of a parallelogram depending on the known data, you can use the following formulas:
- S = a*h, where a is the side of the parallelogram, h is the height drawn to a.
- S = a*b*sinα, where a and b are the sides of the figure, α is the angle between these sides.
After that. How did you figure it out? How to find the area of a parallelogram, you can begin to find the volume of our parallelepiped. To do this we use the formula:
V = S*h, where S is the base area obtained earlier, h is the height of our parallelepiped.
Let's look at an example.
We are given a parallelepiped with a height of 50 cm, the base (parallelogram) of which has a side equal to 23 cm and the height drawn to this side is 8 cm. We substitute the above formula:
S = 23*8 = 184 sq. cm.
Now we substitute the formula to find the volume of a parallelepiped:
V = 184*50 = 9,200 cubic meters
Mathematics lesson ‘Volume of a rectangular parallelepiped’ (5th grade)
Answer: the volume of this parallelepiped is 9200 cubic centimeters.
Third way.
This option is only suitable for rectangular type parallelepiped, sides whose bases will be equal. To do this, you just need to cube these sides.
V = a3, i.e. cubed
Given a parallelepiped with a base side of 12. This means that the volume of this figure is calculated by the following formula V = 123 = 1728 cc cm.
Either method is very simple. The main thing is to arm yourself with a calculator and perform all the calculations correctly. Good luck!
volume of a rectangular parallelepiped
S1*2 + S2*2 + S3*2 = S
Parallelepiped base
The calculator will calculate and write out the solution in detail and with comments. All you have to do is copy the line solution of the parallelepiped into your notebook. A detailed text solution with explanations will allow you to understand the methodology for solving such problems and, if necessary, answer questions by giving a detailed and competent answer.
Calculation of volume and area of a parallelogram is an elementary basis for many technical and everyday calculations!
Volumes. Volume of a rectangular parallelepiped
For example, to calculate repairs in a room, calculate data for heating or air conditioning.
rectangular parallelogram
The formula used in our calculator will find volume of a rectangular parallelepiped. And if your parallelepiped has oblique edges, instead of the length of the corresponding oblique edge, you must enter the value of the height of this part of the figure.
Formula for the volume of a rectangular parallelepiped
To find it, you need to know the dimensions of the ribs: height, width and length. According to the formula, the dimensions of the parallelepiped faces must be multiplied in any order.
The volume can be expressed in liters or cubic cm, cubic millimeters.
Formula for the surface area of a parallelepiped
S1*2 + S2*2 + S3*2 = S
Using the formula for the area of a parallelepiped, you need to find the areas of all sides of the parallelepiped and then add them up. Opposite sides, faces, and edges of a parallelepiped are equal to each other, so when calculating areas, you can use multiplication by two.
Parallelepiped base
In some cases, the base area of the parallelepiped is known, then in order to find the volume it is enough to multiply the base area by the height. ! IMPORTANT! - this is true only for a rectangular parallelepiped.
How to find the volume of a parallelepiped?
The easiest way to find the volume is by entering three known values into columns online calculator volume! Then - press the button - you will get the result)!
The calculator will calculate volume of parallelepiped abcda1b1c1d1 and will describe the decision in detail and with comments.
Volume of a rectangular parallelepiped
All you have to do is copy the line solution of the parallelepiped into your notebook. A detailed text solution with explanations will allow you to understand the methodology for solving such problems and, if necessary, answer questions by giving a detailed and competent answer.
Calculation of volume and area of a parallelogram is an elementary basis for many technical and everyday calculations! For example, to calculate repairs in a room, calculate data for heating or air conditioning.
A parallelogram is a three-dimensional geometric figure that has six sides, each side being a parallelogram. The sides of a parallelogram are usually called faces. If all the faces of a parallelepiped have the shape of a rectangle, then this is already rectangular parallelogram! This figure is designated by the letters abcda1b1c1d1.
School is an immense bowl of knowledge, which includes many disciplines that can interest any child. Mathematics is the queen exact sciences. Strict and disciplined, she does not tolerate inaccuracies. Even as an adult, ordinary life we may encounter different math problems: calculation square meters for laying tiles in the bathroom, cubic meters for determining the volume of a tank, etc., let alone schoolchildren who are just starting their mathematical journey.
Very often, when starting to study mathematics, or more precisely, geometry, students confuse flat figures with three-dimensional ones. A cube is called a square, a ball is called a circle, and a parallelepiped is called an ordinary rectangle. And there are some subtleties here.
It is difficult to help a child complete homework, not knowing exactly whether the volume or area of a figure - flat or volumetric - needs to be found. Volume cannot be found flat figures, such as square, circle, rectangle. In their case, you can only find the area. Before proceeding with the task, you should prepare the necessary attributes:
- A ruler to measure the data we need.
- Calculator for further calculations.
First, let's look at the very concept of a volumetric rectangle. This is a parallelepiped. At its base there is a parallelogram. Since he has six of them, therefore all parallelograms are faces of a parallelepiped.
As for its faces, they may differ, that is, if the straight side faces are rectangles, then this is a right parallelepiped, but if all six faces are rectangles, then we have a rectangular parallelepiped.
- After reading the problem, you need to determine what exactly should be found; length of a figure, volume or area.
- Which part of the figure is considered in the problem - an edge, a vertex, a face, a side, or maybe the whole figure?
Having defined all the assigned tasks, you can proceed directly to the calculations. For this we need special formulas. So, in order to find the volume of a rectangular parallelepiped, the length, width and height (that is, the thickness of the figure) are multiplied together. The formula for calculating the volume of a rectangular parallelepiped is as follows:
V=a*b*h,
V is the volume of the parallelepiped, where a- its length b- width and h- height accordingly.
Important! Before you begin, convert all measurements into one unit of calculation. The answer must certainly be in cubic units.
Example one
Let us determine the volume of the alcohol tank with the following dimensions:
- length three meters;
- width two meters fifty centimeters;
- height three hundred centimeters.
First, be sure to agree on the units of measurement and multiply them:
Multiplying the data, we get the answer in cubic meters, that is, 3*2.5*3= 22.5 meters per cube.
Example two
The cabinet is four meters high, seventy centimeters wide and 80 centimeters deep.
Knowing the calculation formula, you can perform multiplication. But there is no need to rush, as was said at the beginning, the units should be coordinated with each other, that is, if you want to calculate in centimeters, convert all calculations into centimeters, or if in meters, then into meters. Let's do both options.
So, let's start with centimeters. Convert meters to centimeters:
V = 400 * 70 * 80;
V = 2240000 centimeters cubed.
Now meters:
V = 4* 0.7 * 0.8;
V = 2.24 meters cubed.
Based on the above manipulations, it is obvious that working with cubic meters easier and more understandable.
Example three
Given a room, the volume of which must be calculated. The length of this room is five meters, the width is three, and the ceiling height is 2.5. Again we use the formula we know:
V = a * b * h;
where a is the length of the room and is equal to 5, b is the width and is equal to 3 and h is the height, which is equal to 2.5
Since all units are given in meters, you can immediately begin calculations. Multiplying a, b and h together:
V = 5 * 3 * 2.5;
V = 37.5 meters cubed.
So, as a conclusion, we can say that knowing the basic mathematical rules to calculate the volume or area of figures, as well as by correctly identifying figures (flat or volumetric), being able to convert centimeters to meters and vice versa - you can make it easier for your child to study geometry, which cannot but make this process more interesting and attractive, because all the accumulated knowledge at school can be successfully used in the most ordinary everyday life in the future.
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