Application of formulas for the volume and surface area of a rectangular parallelepiped for solving practical problems and mathematical modeling. Application of formulas for the volume and surface area of a rectangular parallelepiped to solve practical problems and ma
The upper (lower) face will be equal to ab, i.e. 7x6=42 cm. The area of one of the side faces will be equal to bc, i.e. 6x4=24 cm. Finally, the area of the front (back) face will be equal to ac, i.e. 7x4=28 cm.
Now add together all three results and multiply the resulting sum by two. In ours it will look like this: 42+24+28=94; 94x2=188. Thus, the surface area of a given cuboid will be equal to 188 cm.
note
Be careful not to confuse a rectangular box with a straight one. For a right parallelepiped, only the sides (4 of 6 faces) are rectangles, and the upper and lower bases are arbitrary parallelograms.
A cube can be considered as a special case of a rectangular parallelepiped. Since all its faces are equal, to find its surface it will be necessary to square the length of the edge and multiply by 6.
Sources:
- An online calculator that calculates the surface area of a cuboid
- how to find a cuboid
A cuboid is a polyhedral figure consisting of six rectangles. Knowing the length of all its faces, you can calculate its volume, diagonal, surface area.
You will need
- The dimensions of the edges of a rectangular parallelepiped.
Instruction
Calculation of the surface area of a rectangular parallelepiped.
Let us be given a rectangular parallelepiped with sides a, b, c. Then, in order to calculate its surface area S, you need to use the formula:
S = 2+(a*b+b*c+a*c)
Parallelepiped - geometric volumetric figure, representing special case quadrangular prism. Like any quadrangular prism, the parallelepiped is a hexagon, but the main distinguishing property parallelepiped is that all its opposite faces are pairwise parallel and equal to each other. In addition to the volume of this figure, the value of its surface area may be of practical interest.
Instruction
The total surface area is the sum of its lateral surface area and its area.
As mentioned above, the opposite faces of the parallelepiped are in pairs between . Therefore, a complete parallelepiped can be defined as twice the sum of the areas of different faces:
S = 2(So + Sb1 + Sb2), where So is the area of the base of the parallelepiped; Sb1, Sb2 are the areas of adjacent side faces of the parallelepiped.
In general, both the bases of a parallelepiped and its side faces are parallelograms. Considering that the area of a parallelogram can be easily found using either of the two formulas below, finding the total surface area of a parallelepiped will not be difficult.
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Useful advice
The area of a parallelogram can be found using any of the following formulas:
1) S = ½ah, where a is the base of the parallelogram; h is its height;
2) S = ½ab∙sinα, where a,b are the lengths of the sides of the parallelogram, α is the acute angle between them.
To solve problems related to determining the surface area of a parallelepiped, it is necessary to clearly understand what a given geometric body, which figures are its side faces and base. Knowledge of the properties of these geometric shapes will help to cope with the solution.
Instruction
A parallelepiped is one that is based on a parallelogram. A parallelogram is a quadrilateral whose opposite sides are equal and parallel. The parallelepiped has an upper and lower base and 4 side faces. All of them are parallelograms. Since the condition does not indicate the angle of inclination of the side faces to the base, it is possible that the prism is straight. This implies a clarification: the side faces of a straight line are rectangles.
In order to find the surfaces of a parallelepiped, you need to find the area of its bases and the area of \u200b\u200bthe lateral surface. To do this, you need to know the length of the sides of the base of the parallelepiped and the length of its edge. To determine the area of the base, you need to draw the height of the parallelogram. We can assume that these values are known, since this item is not specified in the condition. For convenience, the following notations are introduced: AD = BC = a - the bases of the parallelogram; AB = CD = b - the sides of the parallelogram; BN = h - the height of the parallelogram; AE = DL = CK = BF = H - the edge of the parallelepiped.
The area of a parallelogram is defined as the product of its base and its height, i.e. ah. Since the upper and lower bases are equal, their total area is S = 2ah.
Since the side faces are rectangles, their area is calculated as the product of the sides. One side of the face AELD is an edge of the parallelepiped and is equal to H, and the other side of its base is equal to a. Edge area: aH. The lateral faces of the parallelepiped are equal and parallel in pairs. Face AELD is equal to face BFKC. Their total area S = 2aH.
Face AEFB is equal to face DLKC. Side AB coincides with the lateral side of the base of the parallelepiped and is equal to b, side AE is equal to H. Face area AEFB is equal to bH. The sum of the areas of these faces is S = 2bH. Lateral surface of the parallelepiped: 2aH+2bH.
Thus, the total surface area of the parallelepiped is: S = 2ah+2aH+2bH or S = 2(ah+aH+bH) The problem is solved.
A parallelepiped is a prism whose bases and side faces are parallelograms. The parallelepiped can be straight or oblique. How to find its surface area in both cases?
Instruction
The parallelepiped can be straight or oblique. If its edges are perpendicular to the bases, it is straight. The side faces of this are rectangles. At an inclined side faces at an angle to. Its faces are parallelograms. Accordingly, the surfaces of a straight and inclined parallelepiped are defined differently.
The total area of the parallelepiped is the sum of the areas of both bases and its side faces: S=S1+S2.
Determine the area of the base. The area of a parallelogram is equal to the product of its base and its height, i.e. ah. The total area of both bases: S1=2ah.
Determine the area of the side surface of the parallelepiped S1. It is the sum of the areas of all side faces, which are rectangles. The side AD of the face AELD is also the side of the base of the parallelepiped, AD=a. The side LD is its edge, LD=c. The area of a face AELD is equal to the product of its sides, i.e. ac. Opposite faces of the parallelepiped are equal, therefore, AELD=BFKC. Their total area is 2ac.
Side DC of face DLKC is the side side of the base of the box, DC=b. The other side of the face is an edge. Face DLKC is equal to face AEFB. Their total area is 2dc.
Lateral surface area: S2=2ac+2bc. Total surface area of the parallelepiped: S=2ah+2ac+2bc=2(ah+ac+bc).
The difference in finding the surface area of a straight and inclined parallelepiped is that the lateral faces of the latter are also parallelograms, therefore, it is necessary to have their heights. The area of the bases is found in the same way in both cases.
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Parallelepiped - 3D geometric figure with three measuring characteristics: length, width and height. All of them are involved in finding the area of both surfaces of the parallelepiped: full and lateral.
Instruction
A parallelepiped is a polyhedron built on the basis of a parallelogram. It has six faces, which are also these two-dimensional shapes. Depending on how they are located in, a straight and oblique parallelepiped are distinguished. This is expressed in the equality of the angle between the base and the side edge of 90 °.
According to which particular case of the parallelogram the base belongs, one can distinguish a rectangular parallelepiped and its most common variety - a cube. These forms are most commonly found in and are worn standard. They are inherent in household appliances, furniture, electronic devices, etc., as well as in human dwellings themselves, the dimensions of which are great importance for residents and realtors.
Usually, the characteristic is considered to be a set of areas of its faces, the second is the same value plus the areas of both bases, i.e. the sum of all the two-dimensional figures that make up the box. The following formulas are called the main ones along with the volume: Sb \u003d P h, where P is the perimeter of the base, h is the height; Sp \u003d Sb + 2 S, where So is the area of \u200b\u200bthe base.
For special cases, a cube and a figure with rectangular bases, the formulas are simplified. Now it is no longer necessary to determine the height, which is equal to the length of the vertical edge, and the area and perimeter are much easier to find due to the presence of right angles, only length and width are involved in their determination. So, for a rectangular parallelepiped: Sb \u003d 2 s (a + b), where 2 (a + b) is twice the sum of the sides of the base (perimeter), c is the length of the side edge; Sp \u003d Sb + 2 a b \u003d 2 a c + 2 b c + 2 ab = 2 (a c + b c + a b).
In a cube, all edges have the same length, therefore: Sb \u003d 4 a a \u003d 4 a²; Sp \u003d Sb + 2 a² \u003d 6 a².
A parallelepiped is a three-dimensional figure characterized by the presence of edges and edges. Each side face is formed by two parallel side edges and matching sides of both bases. To find side surface parallelepiped, you need to add up the areas of all its vertical or inclined parallelograms.
Instruction
A parallelepiped is a spatial geometric figure that has three: length, height and width. In this regard, he has two horizontal ones, called bases, as well as four lateral ones. All of them have the shape of a parallelogram, but also special cases that simplify not only the graphical representation of the problem, but also the calculations themselves.
The main numerical characteristics of the parallelepiped are the volume. There are full and lateral surfaces of the figure, which are obtained by summing the areas of the corresponding faces, in the first case - all six, in the second - only the lateral ones.
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The purpose of the lesson: In practice, learn to apply the formulas for the volume and surface area of a rectangular parallelepiped.
Tools: multimedia installation, chalk, board, models of parallelepipeds.
During the classes
I. Checking homework.
II. Oral survey.
- How many edges does a cuboid have? What figure are they?
- How many faces does a cuboid have? What figure are they?
- How many vertices does a cuboid have? What figure are they?
III. Work according to ready-made drawings.
- What is a, b and c?
- How to find the area of a side face? Are there other faces with the same area?
- How to find the area of the top face?
- How to find the area of the front face?
- Write on the board the formula for finding the surface area of a parallelepiped.
- Write down the formula for finding the volume of a parallelepiped.
- In what units is the surface area of the parallelepiped measured, and in what units is the volume.
IV. Solve the problem according to the drawing shown in the figure.
Find the surface area and volume of a rectangular parallelepiped.
- 3 * 4 \u003d 12 (sq. cm) - front surface area.
- 3 * 5 \u003d 15 (sq. cm) - lateral surface area.
- 4 * 5 \u003d 20 (sq. cm) - the area of \u200b\u200bthe upper surface.
- 2 * (12 + 15 + 20) \u003d 94 (sq. cm) - the area of \u200b\u200bthe lateral surface of the parallelepiped.
Answer: 94 sq. cm.
V. Practical part. Distribute boxes
- Measure the edges of the parallelepiped (length, height and width). Record the results in a notebook.
- Find the area of the side surface of the parallelepiped.
- Find the volume of the parallelepiped.
- Sign the face of the parallelepiped, the area, which is equal to
- Option 1 - 14 sq. cm
- Option 2 - 18 sq. cm
- Option 3 - 48 sq. cm
VI. Written work on the board with frontal discussion.
Find the surface area and volume of a cuboid with a notch.
- 2*(4*5+5*5+5*4) = 130 square meters cm is the surface area.
- 5*5*4 = 100 cu. cm is the volume of the parallelepiped.
Answer: 130 sq. cm and 100 cu. cm.
VII. A task with practical content.
How many buckets of water, 8 liters each, are poured into the aquarium shown in the figure.
We know that 1 liter = 10 cubic meters.
- 25-5 \u003d 20 (cm) - the height of the poured water.
- 20 * 40 * 60 \u003d 48000 (cubic cm) - the volume of water in the aquarium.
48000 cu. cm = 48 cu. dm = 48 liters - 48:8 = 6 (Ved.) - water will be required.
By the condition of the problem, a rectangular parallelepiped ABCDA 1 B 1 C 1 D 1 is given with dimensions a; b and c:
The task is to find the volume, surface area and sum of the lengths of all the edges of this parallelepiped.
Formula for surface area
The parallelepiped has six faces:
- lower base ABCD;
- top base A 1 B 1 C 1 D 1 ;
- four side faces AA 1 B 1 B; BB 1 C 1 C; CC1D1D; DD 1 A 1 A.
In a cuboid, all faces are rectangles, and the edges are equal:
|AB| = |CD| = |A 1 B 1 | = |C 1 D 1 | = a;
|BC| = |AD| = |B 1 C 1 | = |A 1 D 1 | = b;
|AA 1 | = |BB 1 | = |CC 1 | = |DD 1 | = c.
The sum L of the lengths of all 12 edges is:
L = 4 * a + 4 * b + 4 * c = 4 * (a + b + c);
The surface area of a parallelepiped is the sum of the areas of all six faces. The base areas are the same:
S1 = |AB| *|BC| = |A 1 B 1 | * |B 1 C 1 | = a*b;
The areas of the side faces AA 1 B 1 B and CC 1 D 1 D are the same and equal:
S2 = |AB| * |AA 1 | = |CD| * |CC 1 | = a*c;
The areas of the remaining two faces BB 1 C 1 C and DD 1 A 1 A are also equal:
S3 = |BC| * |BB 1 | = |AD| * |AA 1 | = b*c;
The surface area is:
S = 2 * S1 + 2 * S2 + 2 * S3 = 2 * a * b + 2 * a * c + 2 * b * c = 2 * (a * b + a * c + b * c);
The volume of a rectangular parallelepiped is equal to its three dimensions:
V = S1 * |AA 1 | = a*b*c;
Calculation of required parameters
Substituting the initial data, we get:
L = 4 * (0.24 + 0.4 + 1.5) = 8.56 (m);
S \u003d 2 * (0.24 * 0.4 + 0.24 * 1.5 + 0.4 * 1.5) \u003d 2.112 (m ^ 2);
V \u003d 0.24 * 0.4 * 1.5 \u003d 0.144 (m ^ 3);
Answer: L = 8.56 (m); S = 2.112 (m^2); V = 0.144 (m^3);
one). V \u003d a ∙ b ∙ c - a formula for finding the volume of a rectangular parallelepiped V with base length a, width b and height c. The dimensions of a rectangular parallelepiped are: a = 0.24 m, b = 0.4 m, c = 1.5 m. Then:
V = 0.24 m ∙ 0.4 m ∙ 1.5 m = 0.144 m³.
2). S \u003d 2 ∙ (a ∙ b + a ∙ c + b ∙ c) - the surface area of \u200b\u200bthe parallelepiped is equal to the sum of the areas of all its six faces. We get:
S = 2 ∙ (0.24 m ∙ 0.4 m + 0.24 m ∙ 1.5 m + 0.4 m ∙ 1.5 m) = 2 ∙ (0.096 + 0.36 + 0.6) m² = 2 ∙ 1.056 m² = 2.112 m²
3). L \u003d 4 ∙ (a + b + c) - the sum of the lengths of all twelve edges of the parallelepiped. Means:
L = 4 ∙ (0.24 m + 0.4 m + 1.5 m) = 4 ∙ 2.14 m = 8.56 m.
Answer: 0.144 m³ - volume, 2.112 m² - surface area and 8.56 m - the sum of the lengths of all edges of this rectangular parallelepiped.
