Every high school student is aware of such spatial figures as a ball, cylinder, cone, pyramid and prism. From this article you will learn about what a triangular prism is and what its characteristics are.
What figure will be considered in the article?
A triangular prism is the simplest representative of the class of prisms, which has fewer sides, vertices, and edges than any other similar spatial figure. This prism is formed by two triangles, which can have an arbitrary shape, but which must necessarily be equal to each other and located in parallel planes in space, and three parallelograms that are not equal in the general case. For clarity, the following figure is described.
How can I get a triangular prism? Very simple: you should take a triangle and transfer it to some vector in space. Then connect the identical vertices of two triangles with segments. So we get the frame of the figure. If we now imagine that this frame limits the solid sides, then we get the volumetric figure depicted.
What elements does the prism under study consist of?
A triangular prism is a polyhedron, that is, it is formed by several intersecting faces or sides. It was indicated above that she has five such sides (two triangular and three quadrangular). Triangular sides are called bases, while parallelograms are lateral faces.
Like any polyhedron, the prism under study has vertices. Unlike the pyramid, the vertices of any prism are equal. The triangular figure has six. All of them belong to both grounds. At each vertex, two edges of the base intersect and one side edge.
If we add the number of vertices with the number of sides of the figure, and then subtract the number 2 from the obtained value, we get the answer to the question of how many edges the prism in question has. There are nine of them: six limit the bases, and the remaining three share parallelograms among themselves.
Types of figures
The rather detailed description of a triangular prism given in the previous paragraphs corresponds to several types of figures. Consider their classification.
The prism under study can be inclined and straight. The difference between them is the type of side faces. In a direct prism, they are rectangles, and in an inclined prism they are parallelograms of a general form. Two prisms with triangular bases are shown below, one of which is straight and the other inclined.
Unlike an inclined one, in a direct prism, all angles dihedral between the bases and sides are 90 °. What does the last fact mean? That the height of the triangular prism, that is, the distance between its bases, in a straight figure is equal to the length of any side rib. In an inclined figure, the height is always less than the length of any lateral rib.
A prism with a triangular base may be incorrect and correct. If its bases are triangles with equal sides, and the figure itself is straight, then it is called correct. A regular prism has a fairly high symmetry, including reflection planes and rotation axes. For a correct prism, formulas for calculating its volume and the surface area of faces will be given below. So, in order.
Triangular Prism Area
Before proceeding to obtain the corresponding formula, we present a scan of the correct prism.
It is clear that the area of the figure can be calculated by adding three areas of the same rectangles and two areas of equal triangles with the same sides. We denote the height of the prism by the letter h, and the side of its triangular base by the letter a. Then for the area of the triangle S 3 we have:
S 3 = √3 / 4 * a 2
This expression is obtained if we multiply the height of the triangle by its base, and then divide the result by 2.
For the area of rectangle S 4 we get:
S 4 = a * h
Adding the area of all sides, we get the total surface area of the figure:
S = 2 * S 3 + 3 * S 4 = √3 / 2 * a 2 + 3 * a * h
Here, the first term reflects the area of the bases, and the second is the area of the lateral surface of the triangular prism.
Recall that this formula is valid only for the correct figure. In the case of an incorrect inclined prism, the calculation of the area should be carried out in stages: first determine the area of the bases, and then the lateral surface. The latter will be equal to the product of the side rib and the perimeter perpendicular to the side faces of the slice.
Figure volume
The volume of a triangular prism can be calculated by the formula common to all figures of this class. It has the form:
V = S o * h
In the case of a regular triangular prism, this formula will take the following concrete form:
V = √3 / 4 * a 2 * h
If the prism is incorrect but straight, then the corresponding area for the triangle should be substituted for the base area. If the prism is inclined, then, in addition to determining the area of the base, its height should also be calculated. As a rule, trigonometric formulas are used for this if dihedral angles between the sides and the bases are known.