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Volume and surface area of ​​a regular quadrangular prism


Prism is a three-dimensional figure with two equal parallel bases. The shape at the base determines the type of prism, for example, a rectangular or triangular prism. Since the prism is a three-dimensional figure, it is often necessary to calculate the volume (the space bounded by the side faces and bases) of the prism. But sometimes in tasks it is required to find the height of the prism. It is not so difficult if the necessary information is given: the volume or surface area and the perimeter of the base. The formulas given in this article are applicable to prisms with bases of any shape, if you know how to calculate the area of ​​the base.

What does a prism look like

A regular quadrangular prism is a hexagon, at the base of which there are 2 squares, and the side faces are represented by rectangles. Another name for this geometric figure is a straight parallelepiped.

The figure, which depicts a quadrangular prism, is shown below.

You can also see in the picture the most important elements that make up a geometric body. It is customary to relate to them:

  1. Prism Basics - squares LMNO and L₁M₁N₁O₁.
  2. Side faces - rectangles MM₁L₁L, LL₁O₁O, NN₁O₁O and MM₁N₁N, located at right angles to the bases.
  3. Side ribs - segments located at the junction between two side faces: M₁M, N₁N, O₁O and L₁L. They also play the role of height (since they lie in a plane parallel to the bases). In the prism, the side edges are always equal to each other - this is one of the most important properties of this geometric body.
  4. Diagonals, which, in turn, are divided into 3 more categories. These include 4 base diagonals (MO, N₁L₁), 8 diagonals of the side faces (ML₁, O₁L) and 4 diagonals of a prism, the beginnings and ends of which are vertices of 2 different bases and sides (MOых, N₁L).

Sometimes in geometry problems one can come across the concept of section. The definition will sound like this: a section is all points of a three-dimensional body that belong to a secant plane. The section is perpendicular (crosses the edges of the figure at an angle of 90 degrees). For a rectangular prism, we also consider the diagonal section (the maximum number of sections that can be built is 2) passing through 2 edges and the diagonals of the base.

If the section is drawn so that the cutting plane is not parallel to either the bases or the side faces, the result is a truncated prism.

Various relations and formulas are used to find the reduced prismatic elements. Some of them are known from the planimetric course (for example, to find the base area of ​​a prism, it is enough to recall the square area formula).

Surface area and volume

To determine the volume of a prism by the formula, it is necessary to know the area of ​​its base and height:

Since the base of a regular tetrahedral prism is a square with a side a, you can write the formula in more detail:

If we are talking about a cube - a regular prism with equal length, width and height, the volume is calculated as follows:

To understand how to find the area of ​​the lateral surface of a prism, it is necessary to imagine its development.

From the drawing it can be seen that the side surface is composed of 4 equal rectangles. Its area is calculated as the product of the perimeter of the base and the height of the figure:

Given that the perimeter of the square is P = 4a the formula takes the form:

To calculate the total surface area of ​​the prism, add 2 base areas to the lateral area:

S full = S side + 2 S

In relation to a quadrilateral regular prism, the formula has the form:

For cube surface area:

Knowing the volume or surface area, you can calculate the individual elements of a geometric body.

Finding prism elements

Often there are problems in which the volume is given or the size of the lateral surface area is known, where it is necessary to determine the length of the side of the base or height. In such cases, the formulas can be derived:

  • base side length: a = S side / 4h = √ (V / h),
  • length of height or side rib: h = S side / 4a = V / a²,
  • base area: Sos = V / h,
  • side face area: Side gr = S side / 4.

To determine what area has a diagonal section, you need to know the length of the diagonal and the height of the figure. For square d = a√2. Therefore:

To calculate the diagonal of a prism, use the formula:

To understand how to apply the above relations, you can practice and solve several simple tasks.

Examples of problems with solutions

Here are a few tasks found in state math final exams.

Sand is poured into a box shaped like a regular quadrangular prism. The height of its level is 10 cm. What will the level of sand be if you move it to a container of the same shape, but with a base length 2 times larger?

It should be reasoned as follows. The amount of sand in the first and second containers did not change, i.e., its volume in them coincides. You can indicate the length of the base for a. In this case, for the first box, the volume of the substance will be:

For the second box, the base length is 2abut unknown sand height:

Insofar as V₁ = V₂, you can equate the expressions:

After reducing both sides of the equation by a², we get:

As a result, a new level of sand will be h = 10/4 = 2.5 cm.

ABCDA₁B₁C₁D₁ - the correct prism. It is known that BD = AB₁ = 6√2. Find the area of ​​the full surface of the body.

To make it easier to understand which elements are known, you can draw a figure.

Since we are talking about the correct prism, we can conclude that at the base there is a square with a diagonal of 6√2. The diagonal of the side face has the same size, therefore, the side face also has the shape of a square equal to the base. It turns out that all three dimensions - length, width and height - are equal. We can conclude that ABCDA₁B₁C₁D₁ is a cube.

The length of any edge is determined through the known diagonal:

a = d / √2 = 6√2 / √2 = 6

The total surface area is found by the formula for the cube:

Sful = 6a² = 6 · 6² = 216

The room is being renovated. It is known that its floor has the shape of a square with an area of ​​9 m². The height of the room is 2.5 m. What is the lowest cost of wallpapering a room if 1 m² costs 50 rubles?

Since the floor and ceiling are squares, i.e., regular quadrangles, and its walls are perpendicular to horizontal surfaces, we can conclude that it is a regular prism. It is necessary to determine the area of ​​its lateral surface.

The length of the room is a = √9 = 3 m

Wallpaper will be pasted over the area S lateral = 4 · 3 · 2,5 = 30 m².

The lowest cost of wallpaper for this room is 50·30 = 1500 rubles.

Thus, to solve problems with a rectangular prism, it is enough to be able to calculate the area and perimeter of a square and a rectangle, as well as own the formulas to find the volume and surface area.