The base of a right prism is a triangle. Triangular prism all formulas and example problems
Find all values of a for which the smallest value of the function on the set |x|?1 is not less than ** Equations and inequalities with the GIA Unified State Exam parameter Mathematics Computer science (tasks + solution)
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230. The base of a straight 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.
Face AA1B1B has the maximum area of the side faces.
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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 = S1+S2+S3= 7*5 + 3*5 + 5*5 =75
Sbas= 0.5 * 3 * 5 * sin120=/(4)
Spol=/2
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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 .
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 the area of its bases and the area of the lateral surface.
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 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.
At 10:49 a question was received in the Unified State Exam (school) section, which caused difficulties for the student.
Question that caused difficulties
The base of a straight prism is a triangle with sides 10, 10 and 12. A plane is drawn through the larger 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 Uchis.Ru experts
In order to give a complete answer, a specialist was brought in who is well versed in the required topic of the Unified State Examination (school). Your question was as follows: “The base of a straight prism is a triangle with sides 10, 10 and 12. A plane is drawn through the larger 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 the question you asked will be as follows:
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