In all high school, a stereometry course is conducted, in which the characteristics of various spatial figures are examined. This article is devoted to the study of the properties of one of these figures. Consider what is a regular triangular prism.
Prism in geometry
According to the stereometric definition, a prism is a three-dimensional figure consisting of n parallelograms and two identical n-coal bases, where n is a positive integer. Both bases are located in parallel planes, and parallelograms connect their sides in pairs in a single figure.
Any prism can be obtained in the following way: you should take a flat n-gon and move it parallel to yourself in another plane. In the process of moving the vertices of the n-gon, n segments will be drawn, which will be the side edges of the prism.
Prisms can be convex and concave, straight and oblique, right and wrong. All these types of figures differ from each other in the shape of the n-gons at the base, as well as their location relative to the segment perpendicular to them, the length of which is the height of the prism. The figure below shows a set of prisms with a different number of angles at the base and the number of side faces.
Regular triangular prism
The first prism in the photo above is a regular triangular. It consists of two identical equilateral triangles and of three rectangles. A rectangle is a special case of a parallelogram; therefore, the figure under consideration satisfies the stereometric definition stated above.
In addition to five faces, the triangular prism is formed by six vertices that belong to both bases, and nine edges, three of which are lateral.
An important property of a regular triangular prism is that its height coincides with the length of the side rib. All these ribs are equal to each other, and the side rectangles intersect the bases at right angles. Note that the straight dihedral angles between the bases and side faces lead to the fact that the parallelograms of the inclined prism become rectangles in a straight figure. Obviously, at certain edge lengths, rectangles can become squares.
Important properties of any three-dimensional figure are its surface area and the volume of space enclosed in it. The prism under study is no exception, therefore, we consider its detailed characteristics.
Surface area
The area of a regular triangular prism is formed by the areas of all its five faces. It is known that the area of spatial figures is easier to consider and study on a plane, so it’s convenient to scan the prism. It is shown below.
The scan is represented by five figures of two types, which in the prism were faces.
To determine the area of all these figures, we introduce the following notation: we will consider the length of the side of the base equal to a, and the height (length of the side rib) equal to h. Given the notation, we obtain the area of one triangle:
S 3 = √3 / 4 × a 2
When writing this formula, the standard expression for the area of the triangle was used. The area of one rectangle is:
S 4 = a × h
Given the number of triangles and rectangles (see the scan above), we obtain the formula for the total surface area of the studied geometric figure:
S = 2 × S 3 + 3 × S 4 = √3 / 2 × a 2 + 3 × a × h
Here, the first term on the right side of the equation describes the area of two bases, the second term allows you to calculate the surface area of the side.
Recall that the formula obtained for S is valid only for a direct regular triangular prism. If we were to consider an oblique figure, then the expression for S would have a different form.
Formula for determining the volume of a figure
The volume of any spatial figure is the part of space that is limited by the faces of the polyhedron. The volume of any prism, regardless of the shape of its base and sides, can be determined by the following formula:
V = S 0 × h
That is, it is enough to multiply the area of one base by the height of the whole figure to get the desired value of volume.
For the case of a triangular regular prism, we obtain the following expression for V:
V = S 0 × h = S 3 × h = √3 / 4 × a 2 × h
The written formula for V, as well as the expression for S in the previous paragraph, depend on only two parameters of the figure: lengths a and h. That is, knowledge of just any two linear parameters allows you to calculate all the properties of the studied prism.
The solution of the problem
In physics, a triangular regular prism made of solid glass is often used to decompose the electromagnetic flux in the visible spectrum into a number of frequencies in order to study them. It is necessary to determine how much glass is needed to make a prism with a surface area of 300 cm 2 and a side length of 10 cm.
First we determine the height of the prism h. We use the formula for S, we have:
S = √3 / 2 × a 2 + 3 × a × h =>
h = (S - √3 / 2 × a 2 ) / (3 × a) = (300 - √3 / 2 × 10 2 ) / (3 × 10) = 7.11 cm
Since we know the values of a and h, we use the formula for V to determine the prism volume:
V = √3 / 4 × a 2 × h = √3 / 4 × 10 2 × 7.11 = 307.87 cm 3
Thus, in order to produce the described prism, about 308 cm 3 of glass will be needed.