The base of a right prism is a triangle with sides. The base of a right triangular prism
A triangular prism is a three-dimensional body formed by a combination of rectangles and triangles. In this tutorial, you will learn how to find the size inside (volume) and outside (surface area) of a triangular prism.
triangular prism - this is a pentahedron formed by two parallel planes in which there are two triangles that form two faces of the prism, and the remaining three faces are parallelograms formed by the co-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 side faces of the prism .
The sides of the faces are prism edges(A 1 B 1 , A 1 C 1 , C 1 B 1 , AA 1 , CC 1 , BB 1 , AB, BC, AC), the triangular prism has 9 faces in total.
The height of the prism is the segment of the perpendicular 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. A triangular prism cannot draw such a diagonal.
Base area is the area of the triangular face of the prism.
is the sum of the areas of the quadrilateral faces of the prism.
Types of triangular prisms
There are two types of triangular prism: straight and oblique.
A straight prism has rectangle side faces, while an inclined side face has parallelograms (see fig.)
A prism whose lateral edges are perpendicular to the planes of the bases is called a straight prism.
A prism, the lateral edges of which 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, multiply the area of its base by the height of the prism.
Prism Volume = Base Area x Height
V=S main h
Prism side surface area
To find the lateral surface area of a triangular prism, multiply the perimeter of its base by its height.
Side surface area of a triangular prism = base perimeter x height
S side \u003d P main. h
Total surface area of the prism
To find the area full surface prism, it is necessary to add its base area and the area of \u200b\u200bthe lateral surface.
since S side \u003d P main. h, we get:
S full =P main. h+2S main
Correct prism is a right prism whose base is a regular polygon.
Prism Properties:
The top and bottom bases of a prism are equal polygons.
The side faces of the prism look like a parallelogram.
The side 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 area of the base is in cm2, then the height should be expressed in centimeters and the volume in cm3. 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
— 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.
Tasks for calculating a triangular prism
Task 1. The base of a right triangular prism is 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 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). So the volume is:
V = 1/2 6 8 5 = 120.
Task 2.
A plane parallel to the lateral 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 main h.
The triangle at the base of the original prism is similar to the triangle at the base of the truncated prism. The similarity coefficient is 2, since the section is drawn through the midline (linear dimensions larger triangle twice 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 \u003d S 1 k 2 \u003d S 1 2 2 \u003d 4S 1.
Base area of the entire prism more area bases of a cut-off prism by 4 times. 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 desired volume is 20.
Find all values of a for which smallest value functions on the set |x|?1 not less than ** Equations and inequalities with GIA parameter USE Mathematics Informatics (tasks + solution)
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230. The base of a right prism is a triangle with sides of 5 cm and 3 cm and an angle of 120° between them. The largest area of the side faces is 35 cm2. Find the lateral surface area of the prism.
Let the edge of the prism, that is, its height, be equal to H.
The AA1B1B face has the maximum area of the side faces.
Select it with the mouse and press CTRL + ENTER
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The base of a right prism is a triangle with sides 5 and 3
The base of a right prism is a triangle with sides 5 and 3
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Sside \u003d S1 + S2 + S3 \u003d 7 * 5 + 3 * 5 + 5 * 5 \u003d 75
Sbase= 0.5 * 3 * 5 * sin120=/(4)
Spol=/2
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At 10:49 a question was received at USE section(school), which caused difficulties for the student.
Question causing difficulty
The base of a straight prism is a triangle with sides 10, 10 and 12. A plane is drawn through the large side of the lower base and the middle of the opposite side edge at an angle of 60 ° to the plane of the base. Find the volume of the prism.Answer prepared by experts Learn.Ru
In order to give a full answer, a specialist was involved who is well versed in the required subject "USE (school)". Your question was as follows: "The base of a right prism is a triangle with sides 10, 10 and 12. A plane is drawn through the large side of the lower base and the middle of the opposite side edge at an angle of 60 ° to the plane of the base. Find the volume of the prism."
After a meeting with other specialists of our service, we are inclined to believe that the correct answer to your question will be as follows:
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For you, a few more simple tasks to solve the prism. Consider a right prism with a right triangle at the base. The question is raised about finding the volume or surface area. Prism volume formula:
Prism surface area formula (general):
*At a straight prism side surface consists of rectangles and is equal to the product of the perimeter of the base and the height of the prism. Remember the formula for the area of a triangle. In this case, we have a right triangle - its area is equal to half the product of the legs. Consider the tasks:
The base of a right triangular prism is a right triangle with legs 10 and 15, the side edge is 5. Find the volume of the prism.
The area of the base is the area of a right triangle. It is equal to half the area of a rectangle with sides 10 and 15).
Thus, the desired volume is equal to:
Answer: 375
The base of a right triangular prism is a right triangle with legs 20 and 8. The volume of the prism is 400. Find its side edge.
The problem is the reverse of the previous one.
Prism volume:
The area of the base is the area of a right triangle:
In this way
Answer: 5
The base of a right triangular prism is a right triangle with legs 5 and 12, the height of the prism is 8. Find its surface area.
The surface area of a prism is the sum of the areas of all faces - these are two equal bases and a side surface.
In order to find the areas of all faces, it is necessary to find the third side of the base of the prism (the hypotenuse of a right triangle).
According to the Pythagorean theorem:
Now we can find the base area and lateral surface area. The base area is:
The area of the lateral surface of the prism with the perimeter of the base is equal to:
*You can do without the formula and just add up the areas of three rectangles: