What is the area of the lateral surface of the prism? Prism lateral surface area
IN school curriculum In 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 side faces represented by rectangles. Another name for this geometric figure- straight parallelepiped.
A drawing showing a quadrangular prism is shown below.
You can also see in the picture essential elements, of which the geometric body consists. 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 ( maximum amount sections that can be constructed - 2), passing through 2 edges and 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 new level sand 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
The lateral surface area of the prism. Hello! In this publication we will analyze a group of problems in stereometry. Let's consider a combination of bodies - a prism and a cylinder. On this moment This article completes the entire series of articles related to the consideration of types of tasks in stereometry.
If new ones appear in the task bank, then, of course, there will be additions to the blog in the future. But what is already there is quite enough for you to learn how to solve all the problems with a short answer as part of the exam. There will be enough material for years to come (the mathematics program is static).
The presented tasks involve calculating the area of a prism. I note that below we consider a straight prism (and, accordingly, a straight cylinder).
Without knowing any formulas, we understand that the lateral surface of a prism is all its lateral faces. A straight prism has rectangular side faces.
The area of the lateral surface of such a prism is equal to the sum of the areas of all its lateral faces (that is, rectangles). If we are talking about a regular prism into which a cylinder is inscribed, then it is clear that all the faces of this prism are EQUAL rectangles.
Formally, the lateral surface area of a regular prism can be reflected as follows:
27064. A regular quadrangular prism is circumscribed about a cylinder whose base radius and height are equal to 1. Find the lateral surface area of the prism.
The lateral surface of this prism consists of four rectangles of equal area. The height of the face is 1, the edge of the base of the prism is 2 (these are two radii of the cylinder), therefore the area of the side face is equal to:
Side surface area:
73023. Find the lateral surface area of a regular triangular prism circumscribed about a cylinder whose base radius is √0.12 and height is 3.
The lateral surface area of this prism is equal to the sum three squares side faces (rectangles). To find the area of the side face, you need to know its height and the length of the base edge. The height is three. Let's find the length of the base edge. Consider the projection (top view):
We have a regular triangle into which a circle with radius √0.12 is inscribed. From the right triangle AOC we can find AC. And then AD (AD=2AC). By definition of tangent:
This means AD = 2AC = 1.2. Thus, the lateral surface area is equal to:
27066. Find the lateral surface area of a regular hexagonal prism circumscribed about a cylinder whose base radius is √75 and height is 1.
The required area is equal to the sum of the areas of all side faces. A regular hexagonal prism has lateral faces that are equal rectangles.
To find the area of a face, you need to know its height and the length of the base edge. The height is known, it is equal to 1.
Let's find the length of the base edge. Consider the projection (top view):
We have a regular hexagon into which a circle of radius √75 is inscribed.
Let's consider right triangle ABO. We know the leg OB (this is the radius of the cylinder). We can also determine the angle AOB, it is equal to 300 (triangle AOC is equilateral, OB is a bisector).
Let's use the definition of tangent in a right triangle:
AC = 2AB, since OB is the median, that is, it divides AC in half, which means AC = 10.
Thus, the area of the side face is 1∙10=10 and the area of the side surface is:
76485. Find the lateral surface area of a regular triangular prism inscribed in a cylinder whose base radius is 8√3 and height is 6.
The area of the lateral surface of the specified prism of three equal-sized faces (rectangles). To find the area, you need to know the length of the edge of the base of the prism (we know the height). If we consider the projection (top view), we have a regular triangle inscribed in a circle. The side of this triangle is expressed in terms of radius as:
Details of this relationship. So it will be equal
Then the area of the side face is: 24∙6=144. And the required area:
245354. A regular quadrangular prism is circumscribed about a cylinder whose base radius is 2. The lateral surface area of the prism is 48. Find the height of the cylinder.
Definition. Prism is a polyhedron, all of whose vertices are located in two parallel planes, and in these same two planes lie two faces of the prism, which are equal polygons with correspondingly parallel sides, and all edges that do not lie in these planes are parallel.
Two equal faces are called prism bases(ABCDE, A 1 B 1 C 1 D 1 E 1).
All other faces of the prism are called side faces(AA 1 B 1 B, BB 1 C 1 C, CC 1 D 1 D, DD 1 E 1 E, EE 1 A 1 A).
All side faces form lateral surface prisms .
All lateral faces of the prism are parallelograms .
The edges that do not lie at the bases are called the lateral edges of the prism ( AA 1, BB 1, CC 1, DD 1, EE 1).
Prism diagonal is a segment whose ends are two vertices of a prism that do not lie on the same face (AD 1).
The length of the segment connecting the bases of the prism and perpendicular to both bases at the same time is called prism height .
Designation:ABCDE A 1 B 1 C 1 D 1 E 1. (First, in traversal order, the vertices of one base are indicated, and then, in the same order, the vertices of another; the ends of each side edge are designated by the same letters, only the vertices lying in one base are designated by letters without an index, and in the other - with an index)
The name of the prism is associated with the number of angles in the figure lying at its base, for example, in Figure 1 there is a pentagon at the base, so the prism is called pentagonal prism. But because such a prism has 7 faces, then it heptahedron(2 faces - the bases of the prism, 5 faces - parallelograms, - its side faces)
Among straight prisms, it stands out private view: correct prisms.
A straight prism is called correct, if its bases are regular polygons.
Parallelepiped
Parallelepiped is a quadrangular prism, at the base of which lies a parallelogram (an inclined parallelepiped). Right parallelepiped- a parallelepiped whose lateral edges are perpendicular to the planes of the base.Rectangular parallelepiped- a right parallelepiped whose base is a rectangle.
Properties and theorems:
Some properties of a parallelepiped are similar known properties parallelogram. A rectangular parallelepiped having equal dimensions is called cube
.All faces of a cube are equal squares. The square of the diagonal is equal to the sum of the squares of its three dimensions 
,
where d is the diagonal of the square;
a is the side of the square.
An idea of a prism is given by:
- various architectural structures;
- Kids toys;
- packaging boxes;
- designer items, etc.
The area of the total and lateral surface of the prism
Total surface area of the prism is the sum of the areas of all its faces Lateral surface area is called the sum of the areas of its lateral faces. The bases of the prism are equal polygons, then their areas are equal. That's whyS full = S side + 2S main,
Where S full- total surface area, S side-lateral surface area, S base- base area
The lateral surface area of a straight prism is equal to the product of the perimeter of the base and the height of the prism.
S side= P basic * h,
Where S side-area of the lateral surface of a straight prism,
P main - perimeter of the base of a straight prism,
h is the height of the straight prism, equal to the side edge.
Prism volume
The volume of a prism is equal to the product of the area of the base and the height.
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