Schoolchildren who are preparing to take the Unified State Exam in mathematics should definitely learn how to solve problems on finding the area of a straight and regular prism. Many years of practice confirm the fact that many students consider such geometry tasks to be quite difficult.
At the same time, high school students with any level of training should be able to find the area and volume of a regular and straight prism. Only in this case will they be able to count on receiving competitive scores based on the results of passing the Unified State Exam.
Key Points to Remember
- If the lateral edges of a prism are perpendicular to the base, it is called a straight line. All side faces of this figure are rectangles. The height of a straight prism coincides with its edge.
- A regular prism is one whose side edges are perpendicular to the base in which the regular polygon is located. The side faces of this figure are equal rectangles. A correct prism is always straight.
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Specialists of the Shkolkovo educational project propose to go from simple to complex: first we give theory, basic formulas, theorems and elementary problems with solutions, and then gradually move on to expert-level tasks.
Basic information is systematized and clearly presented in the “Theoretical Information” section. If you have already managed to repeat the necessary material, we recommend that you practice solving problems on finding the area and volume of a right prism. The “Catalogue” section presents a large selection of exercises of varying degrees of difficulty.
Try to calculate the area of a straight and regular prism or right now. Analyze any task. If it does not cause any difficulties, you can safely move on to expert-level exercises. And if certain difficulties do arise, we recommend that you regularly prepare for the Unified State Exam online together with the Shkolkovo mathematical portal, and tasks on the topic “Straight and Regular Prism” will be easy for you.
A triangular prism is a three-dimensional solid formed by connecting 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 base
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 lateral 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 midline 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.
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1. The diagonals of the cube intersect at a point that is the center of the inscribed and circumscribed spheres.
2. The radius of a sphere circumscribed around a cube is equal to .
3. The radius of a sphere inscribed in a cube is equal to 
.
Tasks
1. The diagonal of a cube is . Find its volume.
2. If each edge of a cube is increased by 1, then its surface area will increase by 30. Find the edge of the cube.
3. A ball is inscribed in a cube with edge 6. Find the volume of the sphere divided by .
Answer: 36.
4 . The diagonal of the cube is . Find its volume.
Answer: 27.
5. The diagonal of a cube face is . Find its volume.
6.If each edge of a cube is increased by 1, then its volume will increase by 19. Find the edge of the cube.
7. How many times will the volume of a cube increase if its edges are tripled?
Answer: 27.
8. The diagonal of a cube is 1. Find its surface area.
9. The surface area of a cube is 8. Find its diagonal.
10. The diagonal of a cube's face is 3. Find its surface area.
Answer: 27.
11. The surface area of a cube is 48. Find the diagonal of the cube's face.
12. The diagonal of a cube is . Find its volume.
Answer: 27.
13. The surface area of a cube is 24. Find its volume.
14. How many times will the surface area of a cube increase if its edge is increased three times?
15. The volume of a cube is 27. Find its surface area.
Answer: 54.
16. The volume of a cube is 12. Find the volume of a triangular pyramid cut off from it by a plane passing through the midpoints of two edges emerging from one vertex and parallel to the third edge emerging from the same vertex.
Answer: 1.5.
Rectangular parallelepiped
A parallelepiped is called rectangular if its lateral edges are perpendicular to the base, and the bases are rectangles.
The opposite faces of a rectangular parallelepiped are equal rectangles.
The square of the diagonal of a rectangular parallelepiped is equal to the sum of the squares of its three dimensions 
.
Tasks
1. The diagonal of a rectangular parallelepiped is equal to and forms angles of 30°, 45° and 60° with the planes of the faces of the parallelepiped. Find the volume of the parallelepiped.
Answer: 4.5.
2. A rectangular parallelepiped is circumscribed about a cylinder whose base radius and height are equal to 2. Find the volume of the parallelepiped.
3. Find the volume of the polyhedron shown in the figure, all dihedral angles of which are equal to 90°.
Answer: 7.
4. The volume of a rectangular parallelepiped is equal to 24. One of its edges is equal to 3. Find the area of the face of the parallelepiped perpendicular to this edge.
Answer: 8.
5. The volume of a rectangular parallelepiped is 60. The area of one of its faces is 12. Find the edge of the parallelepiped perpendicular to this face.
Answer: 5.
6. Two edges of a rectangular parallelepiped extending from the same vertex are equal to 2, 4. The diagonal of the parallelepiped is equal to 6. Find the volume of the parallelepiped.
Answer: 32.
7. The edges of a rectangular parallelepiped extending from one vertex are 3, 4, 5. Find its surface area.
Answer: 94.
8. Two edges of a rectangular parallelepiped coming from the same vertex are 3 and 4. The surface area of this parallelepiped is 52. Find the third edge coming from the same vertex.
Answer: 2.
9. Two edges of a rectangular parallelepiped extending from the same vertex are 2, 4. The diagonal of the parallelepiped is 6. Find the surface area of the parallelepiped.
10. Two edges of a rectangular parallelepiped extending from the same vertex are equal to 1, 2. The surface area of the parallelepiped is 16. Find its diagonal.
11. A rectangular parallelepiped is circumscribed about a sphere of radius 2. Find its surface area.
Answer: 96.
12. A rectangular parallelepiped is circumscribed around a sphere of radius 2. Find its volume.
13. The volume of a rectangular parallelepiped circumscribed about a sphere is 216. Find the radius of the sphere.
Answer: 3.
14. The surface area of a rectangular parallelepiped circumscribed around a sphere is 96. Find the radius of the sphere.
Answer: 2.
15. The area of the face of a rectangular parallelepiped is 12. The edge perpendicular to this face is 4. Find the volume of the parallelepiped.
Answer: 48.
16. Two edges of a rectangular parallelepiped coming from the same vertex are 2 and 6. The volume of the parallelepiped is 48. Find the third edge of the parallelepiped coming from the same vertex.
Answer: 4.
17. Two edges of a rectangular parallelepiped coming from one vertex are equal to 2, 3. The volume of the parallelepiped is 36. Find its diagonal.
Answer: 7.
Prism
| prism |
| straight prism |
A polyhedron, two of whose faces are equal polygons lying in parallel planes, and the remaining faces are parallelograms, is called a prism.
Equal polygons lying in parallel planes are called prism bases. The remaining faces are called side faces. They form the lateral surface of the prism. There are ribs at the base and lateral ribs of the prism (L).
A prism is called straight if the lateral edges are perpendicular to the bases of the prism.
The perpendicular dropped from any current of the upper base to the lower base is called the height of the prism (H).
The name of the prism depends on the polygon at the base of the prism.
The total surface of the prism is equal to the sum of the areas of the two bases and the area of the lateral surface.
The lateral surface of the prism is equal to the product of the perimeter of the base and the height of the prism. 
(Or, the product of the perpendicular section perimeter and the lateral edge of the prism 
).
The volume of a prism is equal to the product of the area of the base and the height of the prism.
(Or, the product of the perpendicular cross-sectional area and the lateral edge of the prism 
).
A prism with a parallelogram at its base is called a parallelepiped.
All opposite faces of a parallelepiped are equal and parallel. The diagonals of a parallelepiped intersect at one point and bisect there. The point of intersection of the diagonals is the center of symmetry of the parallelepiped.
A parallelepiped whose all faces are rectangles is called a cuboid.
A rectangular parallelepiped with equal edges is called a cube.
Right prism (triangular regular)
A prism in which the side edges are perpendicular to the bases, and the bases are regular triangles.
1. Side faces - equal rectangles
2. Base side 
Tasks
1. Find the volume of a regular triangular prism, all edges of which are equal.
Answer: 2.25.
2. The volume of a regular triangular prism is 6. What will be the volume of the prism if the sides of its base are tripled and the height is halved?
3. The surface area of a regular triangular prism is 6. What will be the surface area of the prism if all its edges are tripled?
4. 2300 cm3 of water was poured into a vessel shaped like a regular triangular prism and the part was immersed in water. At the same time, the water level rose from 25 cm to 27 cm.
Find the volume of the part. Express your answer in cm3.
5. Water was poured into a vessel shaped like a regular triangular prism. The water level reaches 80 cm. At what height will the water level be if it is poured into another similar vessel, whose base side is 4 times larger than the first? Express your answer in cm.
It is one of the frequent volumetric geometric shapes that we encounter in our lives. For example, on sale you can find keychains and watches in the shape of it. In physics, this figure, made of glass, is used to study the spectrum of light. In this article we will discuss the issue regarding the development of a triangular prism.
What is a triangular prism
Let's look at this figure from a geometric point of view. To obtain it, you should take a triangle with arbitrary side lengths and, parallel to itself, transfer it in space to a certain vector. After this, it is necessary to connect the identical vertices of the original triangle and the triangle obtained by transfer. We got a triangular prism. The photo below shows one example of this figure.
From the figure it can be seen that it is formed by 5 faces. Two identical triangular sides are called bases, three sides represented by parallelograms are called laterals. This prism has 6 vertices and 9 edges, 6 of which lie in the planes of parallel bases.
A triangular prism of the general type was considered above. It will be called correct if the following two mandatory conditions are met:
- Its base must represent a regular triangle, that is, all its angles and sides must be the same (equilateral).
- The angle between each side edge and the base must be straight, that is, 90 o.
The photo above shows the figure in question.
For a regular triangular prism, it is convenient to calculate the length of its diagonals and height, volume and surface area.
Let's take the correct prism shown in the previous figure and mentally carry out the following operations for it:
- Let's first cut the two edges of the upper base, which are closest to us. Bend the base up.
- We will perform the operations of point 1 for the lower base, just bend it down.
- Let's cut the figure along the nearest side edge. Bend two side faces (two rectangles) left and right.
As a result, we will get a scan of a triangular prism, which is presented below.
This scan is convenient to use to calculate the area of the lateral surface and bases of a figure. If the length of the side edge is c, and the length of the side of the triangle is a, then for the area of the two bases we can write the formula:
The area of the lateral surface will be equal to three areas of identical rectangles, that is:
Then the total surface area will be equal to the sum of S o and S b .