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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. At the 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 area of the lateral surface of a given prism is equal to the sum of the areas of the three lateral 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.
Consider the 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.
These are the most common three-dimensional figures among other similar ones that are found in everyday life and nature. Stereometry, or spatial geometry, studies their properties. In this article we will discuss the question of how you can find the lateral surface area of a regular triangular prism, as well as a quadrangular and hexagonal one.
What is a prism?
Before calculating the lateral surface area of a regular triangular prism and other types of this figure, you should understand what they are. Then we will learn to determine the quantities of interest.
A prism, from the point of view of geometry, is a volumetric body that is bounded by two arbitrary identical polygons and n parallelograms, where n is the number of sides of one polygon. It’s easy to draw such a figure; to do this, you should draw some kind of polygon. Then draw a segment from each of its vertices that will be equal in length and parallel to all the others. Then you need to connect the ends of these lines together so that you get another polygon equal to the original one.
Above you can see that the figure is limited by two pentagons (they are called the lower and upper bases of the figure) and five parallelograms, which correspond to rectangles in the figure.
All prisms differ from each other in two main parameters:
- the type of polygon underlying the figure;
- angles between parallelograms and bases.
The number of sides of a rectangle gives the name to a prism. From here we get the above-mentioned triangular, hexagonal and quadrangular figures.
They also differ in the amount of slope. As for the marked angles, if they are equal to 90 o, then such a prism is called straight or rectangular (the angle of inclination is zero). If some of the angles are not right, then the figure is called oblique. The difference between them is clear at first glance. The picture below shows these varieties.
As you can see, the height h coincides with the length of its side edge. In the case of an oblique angle, this parameter is always smaller.
Which prism is called correct?
Since we must answer the question of how to find the lateral surface area of a regular prism (triangular, quadrangular, and so on), we need to define this type of volumetric figure. Let's analyze the material in more detail.
A regular prism is a rectangular figure in which a regular polygon forms identical bases. This figure can be an equilateral triangle, a square, or others. Any n-gon whose side lengths and angles are all the same will be regular.
A number of such prisms are shown schematically in the figure below.
Lateral surface of the prism
As was said in this figure consists of n + 2 planes, which, intersecting, form n + 2 faces. Two of them belong to the bases, the rest are formed by parallelograms. The area of the entire surface consists of the sum of the areas of the indicated faces. If we do not include the values of the two bases, then we get the answer to the question of how to find the lateral surface area of a prism. So, you can determine its meaning and bases separately from each other.
Below is given for which the lateral surface is formed by three quadrangles.
Let's consider the calculation process further. Obviously, the area of the lateral surface of the prism is equal to the sum of the n areas of the corresponding parallelograms. Here n is the number of sides of the polygon forming the base of the figure. The area of each parallelogram can be found by multiplying the length of its side by its height. This applies to the general case.
If the prism under study is straight, then the procedure for determining the area of its lateral surface S b is greatly simplified, since such a surface consists of rectangles. In this case, you can use the following formula:
Where h is the height of the figure, P o is the perimeter of its base
Regular prism and its lateral surface
In the case of such a figure, the formula given in the paragraph above takes on a very specific form. Since the perimeter of an n-gon is equal to the product of the number of its sides and the length of one, the following formula is obtained:
Where a is the side length of the corresponding n-gon.
Lateral surface area of quadrangular and hexagonal
Let's use the formula above to determine the required values for the three types of shapes noted. The calculations will look like this:
For a triangular formula will take the form:
For example, the side of a triangle is 10 cm, and the height of the figure is 7 cm, then:
S 3 b = 3*10*7 = 210 cm 2
In the case of a quadrangular prism, the desired expression takes the form:
If we take the same length values as in the previous example, then we get:
S 4 b = 4*10*7 = 280 cm 2
The lateral surface area of a hexagonal prism is calculated by the formula:
Substituting the same numbers as in the previous cases, we have:
S 6 b = 6*10*7 = 420 cm 2
Note that in the case of a regular prism of any type, its lateral surface is formed by identical rectangles. In the examples above, the area of each of them was a*h = 70 cm 2.
Calculation for an oblique prism
Determining the value of the lateral surface area for a given figure is somewhat more difficult than for a rectangular one. Nevertheless, the above formula remains the same, only instead of the base perimeter, the perpendicular cut perimeter should be taken, and instead of the height, the length of the side edge should be taken.
The picture above shows a quadrangular oblique prism. The shaded parallelogram is the perpendicular slice whose perimeter P sr must be calculated. The length of the side edge in the figure is indicated by the letter C. Then we get the formula:
The perimeter of the cut can be found if the angles of the parallelograms forming the lateral surface are known.
Find the lateral surface area of a regular hexagonal prism, the base side of which is 5 and the height is 10. a H We use the formula for the surface area of a regular prism: At the base lies a regular hexagon, which is divided by large diagonals into 6 equal regular triangles with side a = 5 Therefore, the area of the regular hexagon can be found like this: We use the formula for the area of the lateral surface of a regular prism: a a Substitute the data into the formula * : *
Find the lateral surface area of a regular hexagonal prism, the side of the base of which is 5, and the height is 10. a H The base is a regular hexagon. We use the formula for the lateral surface area of a regular prism: a Substitute the data into the formula * : * S side = = Answer: 300
Find the lateral edge of a regular quadrangular prism if the side of its base is 20 and the surface area is We use the formula for the surface area of a regular prism: At the base there is a square with side a = 20 We use the formula for the lateral surface area of a regular prism: Substitute the data into the formula * : * 1760 = N 1760 = N 80N = N = 12 Answer: 12
Find the volume of a regular hexagonal prism whose base sides are equal to 1 and whose side edges are equal. 3 N a We use the formula for the volume of a regular prism: At the base lies a regular hexagon, which is divided by large diagonals into 6 equal regular triangles with side a = 1 a Therefore, the area of a regular hexagon can be found as follows: H - height (lateral edge) of a regular prism We substitute the data in formula * : *
The face of a parallelepiped is a rhombus with side 1 and an acute angle. One of the edges of the parallelepiped makes an angle of 60 0 with this face and is equal to 2. Find the volume of the parallelepiped.
Through the middle line of the base of a triangular prism, the volume of which is 32, a plane is drawn parallel to the side edge. Find the volume of the cut-off triangular prism. 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.
