Stereometry is a branch of geometry that studies the various properties of figures in the space of a three-dimensional coordinate system. One such figure is a rectangular prism. What it is, and what properties are characteristic of it, we will consider in this article.
Rectangular prism in stereometry
Everyone is familiar with this perfect geometric figure. It is understood as a volumetric object, which consists of six rectangles in the general case, and all of them are pairwise equal. It is not difficult to get this prism in space. It is necessary to take an arbitrary rectangle and move it parallel to itself along a segment perpendicular to the original rectangle. The result is the figure shown below in the figure.
A rectangular prism is also called a box. If its base is a square, then it will become a regular prism, in which the side rectangles will be equal to each other. If at the correct prism the side of the base coincides with the height (the length of the side edge), then we get a cube figure.
Shape Elements
We are talking about the geometric elements that make up the prism under consideration. The first thing that catches your eye when you first look at the figure is its facets. As noted, she has six of them. Two identical faces form the bases of a rectangular prism, the four remaining make up its lateral surface. All faces are either rectangles or squares.
The next important element of the figure is the ribs. Prism has 12 ribs, 8 of which belong to the bases. The remaining four ribs are lateral. Their length is equal to the height of the figure.
Finally, the third important element of the prism under study is its peaks. Unlike a pyramid or cone, a prism does not have a distinguished peak. All of them are her equal. Their number is eight.
As can be seen from the presented quantitative characteristics of the elements of a straight rectangular prism, Euler's theorem holds for their numbers:
number of edges = number of sides + number of vertices - 2 =>
12 = 6 + 8 - 2.
Diagonal shapes
Diagonals of a rectangular prism are of two types:
- those that are located in the plane of the faces of the figure;
- those that are in volume.
If the letters a, b, and h denote the lengths of the sides of the base and the length of the side rib, respectively, then for the length of the diagonals of the first type we can write the following equalities:
d 1 = √ (a 2 + b 2 );
d 2 = √ (a 2 + h 2 );
d 3 = √ (h 2 + b 2 ).
The diagonal d 1 belongs to the bases, and the diagonals d 2 and d 3 lie in the planes of the side rectangles. Obviously, the written formulas follow from the Pythagorean theorem.
As for the diagonals of the second type (volumetric), then any rectangular prism has four such diagonals. Nevertheless, their lengths are equal to each other. The formula for determining the length of the volume diagonal is written as follows:
d 4 = √ (a 2 + b 2 + h 2 ).
If we calculate the diagonal d 4 for the cube, then we can write the following expression, which is obtained from the previous one:
d 4 = a * √3.
In this case, all the diagonals of the faces of the cube will be equal to each other, and their lengths are calculated as follows:
d 1 = d 2 = d 3 = a * √2.
The length of the volume diagonal is always greater than the lengths of the diagonals of the sides.
Determination of surface area
Each student knows that to conveniently determine the surface area that any three-dimensional figure possesses, you should scan it on a plane. A rectangular prism is no exception. Its sweep is simple to do, for this you should cut off two bases from the figure, and then cut it along one of the side ribs. By expanding the edges of the side surface, we get the following picture.
Scan consists of six rectangles of three types. We denote the sides of the base by the letters a and b. The height of the figure is denoted by h. Then the area of one base will be equal to:
S o = a * b
The areas of two different lateral faces are equal:
S 1 = a * h;
S 2 = b * h.
Since the box has a pair of identical faces, the area formulas for which are written, the total surface area of the figure S will be equal to:
S = 2 * (S o + S 1 + S 2 ) = 2 * (a * b + a * h + b * h).
The formula for S can be simplified if the rectangular prism has additional symmetry. For example, if the sides of its base are equal (a = b), then for S we can write the expression:
S = 2 * a * (a + 2 * h).
This expression follows from the previous formula. Accordingly, if the height and length of the base are equal (h = a), then we get a cube whose surface area is:
S = 6 * a 2 .
Note the higher the symmetry of the parallelepiped, the smaller the number of linear parameters you need to know in order to calculate the value S.
Rectangular prism volume
The studied figure consists of six faces that limit the spatial volume. It is the volume of the figure itself. To calculate it, you can apply the universal formula for all prisms and cylinders. It has the following form:
V = S o * h
Since the base of the studied figure is a rectangle, and its height is equal to the length of the side edge, the volume of the rectangular prism will be equal to:
V = a * b * h
It is also useful to give formulas for a regular prism with a square base and for a cube, their volumes are calculated as follows:
for the correct prism: V = a 2 * h;
for a cube: V = a 3 .
As for the area, to determine the volume, you need to know from 1 to 3 linear parameters depending on the symmetry of the box.