Edges of a regular quadrangular prism. Triangular prism all formulas and example problems
Maintaining your privacy is important to us. For this reason, we have developed a Privacy Policy that describes how we use and store your information. Please review our privacy practices and let us know if you have any questions.
Collection and use of personal information
Personal information refers to data that can be used to identify or contact a specific person.
You may be asked to provide your personal information at any time when you contact us.
Below are some examples of the types of personal information we may collect and how we may use such information.
What personal information do we collect:
- When you submit an application on the site, we may collect various information, including your name, telephone number, email address, etc.
How we use your personal information:
- The personal information we collect allows us to contact you with unique offers, promotions and other events and upcoming events.
- From time to time, we may use your personal information to send important notices and communications.
- We may also use personal information for internal purposes, such as conducting audits, data analysis and various research in order to improve the services we provide and provide you with recommendations regarding our services.
- If you participate in a prize draw, contest or similar promotion, we may use the information you provide to administer such programs.
Disclosure of information to third parties
We do not disclose the information received from you to third parties.
Exceptions:
- If necessary - in accordance with the law, judicial procedure, in legal proceedings, and/or on the basis of public requests or requests from government bodies in the Russian Federation - to disclose your personal information. We may also disclose information about you if we determine that such disclosure is necessary or appropriate for security, law enforcement, or other public importance purposes.
- In the event of a reorganization, merger, or sale, we may transfer the personal information we collect to the applicable successor third party.
Protection of personal information
We take precautions - including administrative, technical and physical - to protect your personal information from loss, theft, and misuse, as well as unauthorized access, disclosure, alteration and destruction.
Respecting your privacy at the company level
To ensure that your personal information is secure, we communicate privacy and security standards to our employees and strictly enforce privacy practices.
In the school curriculum for a stereometry course, the study of three-dimensional figures usually begins with a simple geometric body - the polyhedron of a prism. The role of its bases is performed by 2 equal polygons lying in parallel planes. A special case is a regular quadrangular prism. Its bases are 2 identical regular quadrangles, to which the sides are perpendicular, having the shape of parallelograms (or rectangles, if the prism is not inclined).
What does a prism look like?
A regular quadrangular prism is a hexagon, the bases of which are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure is a straight parallelepiped.
A drawing showing a quadrangular prism is shown below.
You can also see in the picture the most important elements that make up a geometric body. These include:
Sometimes in geometry problems you can come across the concept of a section. The definition will sound like this: a section is all the points of a volumetric body belonging to a cutting plane. The section can be perpendicular (intersects the edges of the figure at an angle of 90 degrees). For a rectangular prism, a diagonal section is also considered (the maximum number of sections that can be constructed is 2), passing through 2 edges and the diagonals of the base.
If the section is drawn in such a way that the cutting plane is not parallel to either the bases or the side faces, the result is a truncated prism.
To find the reduced prismatic elements, various relations and formulas are used. Some of them are known from the planimetry course (for example, to find the area of the base of a prism, it is enough to recall the formula for the area of a square).
Surface area and volume
To determine the volume of a prism using the formula, you need to know the area of its base and height:
V = Sbas h
Since the base of a regular tetrahedral prism is a square with side a, You can write the formula in more detailed form:
V = a²·h
If we are talking about a cube - a regular prism with equal length, width and height, the volume is calculated as follows:
To understand how to find the lateral surface area of a prism, you need to imagine its development.
From the drawing it can be seen that the side surface is made up of 4 equal rectangles. Its area is calculated as the product of the perimeter of the base and the height of the figure:
Sside = Posn h
Taking into account that the perimeter of the square is equal to P = 4a, the formula takes the form:
Sside = 4a h
For cube:
Sside = 4a²
To calculate the total surface area of the prism, you need to add 2 base areas to the lateral area:
Sfull = Sside + 2Smain
In relation to a quadrangular regular prism, the formula looks like:
Stotal = 4a h + 2a²
For the surface area of a cube:
Sfull = 6a²
Knowing the volume or surface area, you can calculate the individual elements of a geometric body.
Finding prism elements
Often there are problems in which the volume is given or the value of the lateral surface area is known, where it is necessary to determine the length of the side of the base or the height. In such cases, the formulas can be derived:
- base side length: a = Sside / 4h = √(V / h);
- height or side rib length: h = Sside / 4a = V / a²;
- base area: Sbas = V / h;
- side face area: Side gr = Sside / 4.
To determine how much area the diagonal section has, you need to know the length of the diagonal and the height of the figure. For a square d = a√2. Therefore:
Sdiag = ah√2
To calculate the diagonal of a prism, use the formula:
dprize = √(2a² + h²)
To understand how to apply the given relationships, you can practice and solve several simple tasks.
Examples of problems with solutions
Here are some tasks found on state final exams in mathematics.
Exercise 1.
Sand is poured into a box shaped like a regular quadrangular prism. The height of its level is 10 cm. What will the sand level be if you move it into a container of the same shape, but with a base twice as long?
It should be reasoned as follows. The amount of sand in the first and second containers did not change, i.e. its volume in them is the same. You can denote the length of the base by a. In this case, for the first box the volume of the substance will be:
V₁ = ha² = 10a²
For the second box, the length of the base is 2a, but the height of the sand level is unknown:
V₂ = h (2a)² = 4ha²
Because the V₁ = V₂, we can equate the expressions:
10a² = 4ha²
After reducing both sides of the equation by a², we get:
As a result, the new sand level will be h = 10 / 4 = 2.5 cm.
Task 2.
ABCDA₁B₁C₁D₁ is a correct prism. It is known that BD = AB₁ = 6√2. Find the total surface area of the body.
To make it easier to understand which elements are known, you can draw a figure.
Since we are talking about a regular prism, we can conclude that at the base there is a square with a diagonal of 6√2. The diagonal of the side face has the same size, therefore, the side face also has the shape of a square equal to the base. It turns out that all three dimensions - length, width and height - are equal. We can conclude that ABCDA₁B₁C₁D₁ is a cube.
The length of any edge is determined through a known diagonal:
a = d / √2 = 6√2 / √2 = 6
The total surface area is found using the formula for a cube:
Sfull = 6a² = 6 6² = 216
Task 3.
The room is being renovated. It is known that its floor has the shape of a square with an area of 9 m². The height of the room is 2.5 m. What is the lowest cost of wallpapering a room if 1 m² costs 50 rubles?
Since the floor and ceiling are squares, i.e. regular quadrangles, and its walls are perpendicular to horizontal surfaces, we can conclude that it is a regular prism. It is necessary to determine the area of its lateral surface.
The length of the room is a = √9 = 3 m.
The area will be covered with wallpaper Sside = 4 3 2.5 = 30 m².
The lowest cost of wallpaper for this room will be 50·30 = 1500 rubles
Thus, to solve problems involving a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and rectangle, as well as to know the formulas for finding the volume and surface area.
How to find the area of a cube
Definition.
This is a hexagon, the bases of which are two equal squares, and the side faces are equal rectangles
Side rib- is the common side of two adjacent side faces
Prism height- this is a segment perpendicular to the bases of the prism
Prism diagonal- a segment connecting two vertices of the bases that do not belong to the same face
Diagonal plane- a plane that passes through the diagonal of the prism and its lateral edges
Diagonal section- the boundaries of the intersection of the prism and the diagonal plane. The diagonal cross section of a regular quadrangular prism is a rectangle
Perpendicular section (orthogonal section)- this is the intersection of a prism and a plane drawn perpendicular to its lateral edges
Elements of a regular quadrangular prism
The figure shows two regular quadrangular prisms, which are indicated by the corresponding letters:
- The bases ABCD and A 1 B 1 C 1 D 1 are equal and parallel to each other
- Side faces AA 1 D 1 D, AA 1 B 1 B, BB 1 C 1 C and CC 1 D 1 D, each of which is a rectangle
- Lateral surface - the sum of the areas of all lateral faces of the prism
- Total surface - the sum of the areas of all bases and side faces (sum of the area of the side surface and bases)
- Side ribs AA 1, BB 1, CC 1 and DD 1.
- Diagonal B 1 D
- Base diagonal BD
- Diagonal section BB 1 D 1 D
- Perpendicular section A 2 B 2 C 2 D 2.
Properties of a regular quadrangular prism
- The bases are two equal squares
- The bases are parallel to each other
- The side faces are rectangles
- The side edges are equal to each other
- Side faces are perpendicular to the bases
- The lateral ribs are parallel to each other and equal
- Perpendicular section perpendicular to all side ribs and parallel to the bases
- Angles of perpendicular section - straight
- The diagonal cross section of a regular quadrangular prism is a rectangle
- Perpendicular (orthogonal section) parallel to the bases
Formulas for a regular quadrangular prism
Instructions for solving problems
When solving problems on the topic " regular quadrangular prism" means that:Correct prism- a prism at the base of which lies a regular polygon, and the side edges are perpendicular to the planes of the base. That is, a regular quadrangular prism contains at its base square. (see properties of a regular quadrangular prism above) Note. This is part of a lesson with geometry problems (section stereometry - prism). Here are problems that are difficult to solve. If you need to solve a geometry problem that is not here, write about it in the forum. To denote the action of extracting the square root in solving problems, the symbol is used√ .
Task.
In a regular quadrangular prism, the base area is 144 cm 2 and the height is 14 cm. Find the diagonal of the prism and the total surface area.Solution.
A regular quadrilateral is a square.
Accordingly, the side of the base will be equal
From where the diagonal of the base of a regular rectangular prism will be equal to
√(12 2 + 12 2 ) = √288 = 12√2
The diagonal of a regular prism forms a right triangle with the diagonal of the base and the height of the prism. Accordingly, according to the Pythagorean theorem, the diagonal of a given regular quadrangular prism will be equal to:
√((12√2) 2 + 14 2 ) = 22 cm
Answer: 22 cm
Task
Determine the total surface of a regular quadrangular prism if its diagonal is 5 cm and the diagonal of its side face is 4 cm.Solution.
Since the base of a regular quadrangular prism is a square, we find the side of the base (denoted as a) using the Pythagorean theorem:
A 2 + a 2 = 5 2
2a 2 = 25
a = √12.5
The height of the side face (denoted as h) will then be equal to:
H 2 + 12.5 = 4 2
h 2 + 12.5 = 16
h 2 = 3.5
h = √3.5
The total surface area will be equal to the sum of the lateral surface area and twice the base area
S = 2a 2 + 4ah
S = 25 + 4√12.5 * √3.5
S = 25 + 4√43.75
S = 25 + 4√(175/4)
S = 25 + 4√(7*25/4)
S = 25 + 10√7 ≈ 51.46 cm 2.
Answer: 25 + 10√7 ≈ 51.46 cm 2.
A triangular prism is a three-dimensional solid formed by combining rectangles and triangles. In this lesson you will learn how to find the size of the inside (volume) and outside (surface area) of a triangular prism.
Triangular prism is a pentahedron formed by two parallel planes in which two triangles are located, forming two faces of a prism, and the remaining three faces are parallelograms formed from the sides of the triangles.
Elements of a triangular prism
Triangles ABC and A 1 B 1 C 1 are prism bases .
The quadrilaterals A 1 B 1 BA, B 1 BCC 1 and A 1 C 1 CA are lateral faces of the prism .
The sides of the faces are prism ribs(A 1 B 1, A 1 C 1, C 1 B 1, AA 1, CC 1, BB 1, AB, BC, AC), a triangular prism has 9 faces in total.
The height of a prism is the perpendicular segment that connects the two faces of the prism (in the figure it is h).
The diagonal of a prism is a segment that has ends at two vertices of the prism that do not belong to the same face. For a triangular prism such a diagonal cannot be drawn.
Base area is the area of the triangular face of the prism.
is the sum of the areas of the quadrangular faces of the prism.
Types of triangular prisms
There are two types of triangular prism: straight and inclined.
A straight prism has rectangular side faces, and an inclined prism has parallelogram side faces (see figure)
A prism whose side edges are perpendicular to the planes of the bases is called a straight line.
A prism whose side edges are inclined to the planes of the bases is called inclined.
Basic formulas for calculating a triangular prism
Volume of a triangular prism
To find the volume of a triangular prism, you need to multiply the area of its base by the height of the prism.
Prism volume = base area x height
V=S basic h
Prism lateral surface area
To find the lateral surface area of a triangular prism, you need to multiply the perimeter of its base by its height.
Lateral surface area of a triangular prism = base perimeter x height
S side = P main h
Total surface area of the prism
To find the total surface area of a prism, you need to add its base area and lateral surface area.
since S side = P main. h, then we get:
S full turn =P basic h+2S basic
Correct prism - a straight prism whose base is a regular polygon.
Prism properties:
The upper and lower bases of the prism are equal polygons.
The side faces of the prism have the shape of a parallelogram.
The lateral edges of the prism are parallel and equal.
Tip: When calculating a triangular prism, you must pay attention to the units used. For example, if the base area is indicated in cm 2, then the height should be expressed in centimeters and the volume in cm 3. If the base area is in mm 2, then the height should be expressed in mm, and the volume in mm 3, etc.
Prism example
In this example:
— ABC and DEF make up the triangular bases of the prism
- ABED, BCFE and ACFD are rectangular side faces
— The side edges DA, EB and FC correspond to the height of the prism.
— Points A, B, C, D, E, F are the vertices of the prism.
Problems for calculating a triangular prism
Problem 1. The base of a right triangular prism is a right triangle with legs 6 and 8, the side edge is 5. Find the volume of the prism.
Solution: The volume of a straight prism is equal to V = Sh, where S is the area of the base and h is the side edge. The area of the base in this case is the area of a right triangle (its area is equal to half the area of a rectangle with sides 6 and 8). Thus, the volume is equal to:
V = 1/2 6 8 5 = 120.
Task 2.
A plane parallel to the side edge is drawn through the middle line of the base of the triangular prism. The volume of the cut-off triangular prism is 5. Find the volume of the original prism.
Solution:
The volume of the prism is equal to the product of the area of the base and the height: V = S base h.
The triangle lying at the base of the original prism is similar to the triangle lying at the base of the cut-off prism. The similarity coefficient is 2, since the section is drawn through the middle line (the linear dimensions of the larger triangle are twice as large as the linear dimensions of the smaller one). It is known that the areas of similar figures are related as the square of the similarity coefficient, that is, S 2 = S 1 k 2 = S 1 2 2 = 4S 1 .
The base area of the entire prism is 4 times greater than the base area of the cut-off prism. The heights of both prisms are the same, so the volume of the entire prism is 4 times the volume of the cut-off prism.
Thus, the required volume is 20.
