Different prisms are different from each other. At the same time, they have a lot in common. To find the area of the base of the prism, you will need to understand what type it has.
General theory
A prism is any polyhedron whose sides have the shape of a parallelogram. Moreover, its base can be any polyhedron - from a triangle to an n-gon. Moreover, the bases of the prism are always equal to each other. What does not apply to the side faces is that they can vary significantly in size.
When solving problems, not only the area of the base of the prism is encountered. It may require knowledge of the lateral surface, that is, all the faces that are not bases. Full surface there will already be a union of all the faces that make up the prism.
Sometimes problems involve height. It is perpendicular to the bases. The diagonal of a polyhedron is a segment that connects in pairs any two vertices that do not belong to the same face.
It should be noted that the base area of a straight or inclined prism does not depend on the angle between them and the side faces. If they have the same figures on the top and bottom faces, then their areas will be equal.
Triangular prism
It has at its base a figure with three vertices, that is, a triangle. As you know, it can be different. If so, it is enough to remember that its area is determined by half the product of the legs.
The mathematical notation looks like this: S = ½ av.
To find out the area of the base in general view, the formulas will be useful: Heron and the one in which half of the side is taken to the height drawn to it.
The first formula should be written as follows: S = √(р (р-а) (р-в) (р-с)). This notation contains a semi-perimeter (p), that is, the sum of three sides divided by two.
Second: S = ½ n a * a.
If you need to know the area of the base triangular prism, which is regular, then the triangle turns out to be equilateral. There is a formula for it: S = ¼ a 2 * √3.
Quadrangular prism
Its base is any of the known quadrangles. It can be a rectangle or square, parallelepiped or rhombus. In each case, in order to calculate the area of the base of the prism, you will need your own formula.
If the base is a rectangle, then its area is determined as follows: S = ab, where a, b are the sides of the rectangle.
When we're talking about about a quadrangular prism, then the area of the base correct prism calculated using the formula for a square. Because it is he who lies at the foundation. S = a 2.
In the case when the base is a parallelepiped, the following equality will be needed: S = a * n a. It happens that the side of a parallelepiped and one of the angles are given. Then, to calculate the height, you will need to use an additional formula: n a = b * sin A. Moreover, angle A is adjacent to side “b”, and height n is opposite to this angle.
If there is a rhombus at the base of the prism, then to determine its area you will need the same formula as for a parallelogram (since it is a special case of it). But you can also use this: S = ½ d 1 d 2. Here d 1 and d 2 are two diagonals of the rhombus.
Regular pentagonal prism
This case involves dividing the polygon into triangles, the areas of which are easier to find out. Although it happens that figures can have a different number of vertices.
Since the base of the prism is a regular pentagon, it can be divided into five equilateral triangles. Then the area of the base of the prism is equal to the area of one such triangle (the formula can be seen above), multiplied by five.
Regular hexagonal prism
Using the principle described for a pentagonal prism, it is possible to divide the hexagon of the base into 6 equilateral triangles. The formula for the base area of such a prism is similar to the previous one. Only it should be multiplied by six.
The formula will look like this: S = 3/2 a 2 * √3.
Tasks
No. 1. Given a regular straight line, its diagonal is 22 cm, the height of the polyhedron is 14 cm. Calculate the area of the base of the prism and the entire surface.
Solution. The base of the prism is a square, but its side is unknown. You can find its value from the diagonal of the square (x), which is related to the diagonal of the prism (d) and its height (h). x 2 = d 2 - n 2. On the other hand, this segment “x” is the hypotenuse in a triangle whose legs are equal to the side of the square. That is, x 2 = a 2 + a 2. Thus it turns out that a 2 = (d 2 - n 2)/2.
Substitute the number 22 instead of d, and replace “n” with its value - 14, it turns out that the side of the square is 12 cm. Now just find out the area of the base: 12 * 12 = 144 cm 2.
To find out the area of the entire surface, you need to add twice the base area and quadruple the side area. The latter can be easily found using the formula for a rectangle: multiply the height of the polyhedron and the side of the base. That is, 14 and 12, this number will be equal to 168 cm 2. total area The surface of the prism turns out to be 960 cm 2.
Answer. The area of the base of the prism is 144 cm 2. The entire surface is 960 cm 2.
No. 2. Given At the base there is a triangle with a side of 6 cm. In this case, the diagonal of the side face is 10 cm. Calculate the areas: the base and the side surface.
Solution. Since the prism is regular, its base is an equilateral triangle. Therefore, its area turns out to be equal to 6 squared, multiplied by ¼ and the square root of 3. A simple calculation leads to the result: 9√3 cm 2. This is the area of one base of the prism.
All side faces are the same and are rectangles with sides of 6 and 10 cm. To calculate their areas, just multiply these numbers. Then multiply them by three, because the prism has exactly that many side faces. Then the area of the lateral surface of the wound turns out to be 180 cm 2.
Answer. Areas: base - 9√3 cm 2, lateral surface of the prism - 180 cm 2.
In physics, a triangular prism made of glass is often used to study the spectrum of white light because it can resolve it into its individual components. In this article we will consider the volume formula
What is a triangular prism?
Before giving the volume formula, let's consider the properties of this figure.
To get this, you need to take a triangle of any shape and move it parallel to itself to some distance. The vertices of the triangle in the initial and final positions should be connected by straight segments. The resulting volumetric figure is called a triangular prism. It consists of five sides. Two of them are called bases: they are parallel and equal to each other. The bases of the prism in question are triangles. The three remaining sides are parallelograms.
In addition to the sides, the prism in question is characterized by six vertices (three for each base) and nine edges (6 edges lie in the planes of the bases and 3 edges are formed by the intersection of the sides). If the side edges are perpendicular to the bases, then such a prism is called rectangular.
The difference between a triangular prism and all other figures of this class is that it is always convex (four-, five-, ..., n-gonal prisms can also be concave).
This is a rectangular figure with an equilateral triangle at its base.
Volume of a general triangular prism
How to find the volume of a triangular prism? The formula in general is similar to that for a prism of any type. It has the following mathematical notation:
Here h is the height of the figure, that is, the distance between its bases, S o is the area of the triangle.
The value of S o can be found if some parameters for the triangle are known, for example, one side and two angles or two sides and one angle. The area of a triangle is equal to half the product of its height and the length of the side by which this height is lowered.
As for the height h of the figure, it is easiest to find it for a rectangular prism. In the latter case, h coincides with the length of the side edge.
Volume of a regular triangular prism
General formula volume of a triangular prism, which is given in previous section article can be used to calculate the corresponding value for a regular triangular prism. Since its base is an equilateral triangle, its area is equal to:
Anyone can get this formula if they remember that in equilateral triangle all angles are equal to each other and amount to 60 o. Here the symbol a is the length of the side of the triangle.
The height h is the length of the edge. It is in no way connected with the base of a regular prism and can take arbitrary values. As a result, the formula for the volume of a triangular prism of the correct type looks like this:
Having calculated the root, you can rewrite this formula as follows:
Thus, to find the volume of a regular prism with a triangular base, it is necessary to square the side of the base, multiply this value by the height and multiply the resulting value by 0.433.
Job type: 8
Theme: Prism
Condition
In a regular triangular prism ABCA_1B_1C_1, the sides of the base are 4 and the side edges are 10. Find the cross-sectional area of the prism by the plane passing through the midpoints of the edges AB, AC, A_1B_1 and A_1C_1.
Show solutionSolution
Consider the following figure.
The segment MN is midline triangle A_1B_1C_1, therefore MN = \frac12 B_1C_1=2. Likewise, KL=\frac12BC=2. In addition, MK = NL = 10. It follows that the quadrilateral MNLK is a parallelogram. Since MK\parallel AA_1, then MK\perp ABC and MK\perp KL. Therefore, the quadrilateral MNLK is a rectangle. S_(MNLK) = MK\cdot KL = 10\cdot 2 = 20.
Answer
Job type: 8
Theme: Prism
Condition
The volume of a regular quadrangular prism ABCDA_1B_1C_1D_1 is 24 . Point K is the middle of edge CC_1. Find the volume of the pyramid KBCD.
Solution
According to the condition, KC is the height of the pyramid KBCD. CC_1 is the height of the prism ABCDA_1B_1C_1D_1 .
Since K is the midpoint of CC_1, then KC=\frac12CC_1. Let CC_1=H , then KC=\frac12H. Note also that S_(BCD)=\frac12S_(ABCD). Then, V_(KBCD)= \frac13S_(BCD)\cdot\frac(H)(2)= \frac13\cdot\frac12S_(ABCD)\cdot\frac(H)(2)= \frac(1)(12)\cdot S_(ABCD)\cdot H= \frac(1)(12)V_(ABCDA_1B_1C_1D_1). Hence, V_(KBCD)=\frac(1)(12)\cdot24=2.
Answer
Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.
Job type: 8
Theme: Prism
Condition
Find the lateral surface area of a regular hexagonal prism whose base side is 6 and height is 8.
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Solution
The area of the lateral surface of the prism is found by the formula S side. = P basic · h = 6a\cdot h, where P basic. and h are, respectively, the perimeter of the base and the height of the prism, equal to 8, and a is the side of a regular hexagon, equal to 6. Therefore, S side. = 6\cdot 6\cdot 8 = 288.
Answer
Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.
Job type: 8
Theme: Prism
Condition
Water was poured into a vessel shaped like a regular triangular prism. The water level reaches 40 cm. At what height will the water level be if it is poured into another vessel of the same shape, whose side of the base is twice as large as the first? Express your answer in centimeters.
Solution
Let a be the side of the base of the first vessel, then 2 a is the side of the base of the second vessel. By condition, the volume of liquid V in the first and second vessels is the same. Let us denote by H the level to which the liquid has risen in the second vessel. Then V= \frac12\cdot a^2\cdot\sin60^(\circ)\cdot40= \frac(a^2\sqrt3)(4)\cdot40, And, V=\frac((2a)^2\sqrt3)(4)\cdot H. From here \frac(a^2\sqrt3)(4)\cdot40=\frac((2a)^2\sqrt3)(4)\cdot H, 40=4H, H=10.
Answer
Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.
Job type: 8
Theme: Prism
Condition
In a regular hexagonal prism ABCDEFA_1B_1C_1D_1E_1F_1 all edges are equal to 2. Find the distance between points A and E_1.
Show solutionSolution
Triangle AEE_1 is rectangular, since edge EE_1 is perpendicular to the plane of the base of the prism, angle AEE_1 will be a right angle.
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Then, by the Pythagorean theorem, AE_1^2 = AE^2 + EE_1^2. Let's find AE from triangle AFE using the cosine theorem. Each interior angle of a regular hexagon is 120^(\circ). Then AE^2= AF^2+FE^2-2\cdot AF\cdot FE\cdot\cos120^(\circ)= 2^2+2^2-2\cdot2\cdot2\cdot\left (-\frac12 \right).
Hence, AE^2=4+4+4=12,
AE_1^2=12+4=16,
AE_1=4.
Answer
Source: “Mathematics. Preparation for the Unified State Exam 2017. Profile level." Ed. F. F. Lysenko, S. Yu. Kulabukhova.
Job type: 8
Theme: Prism
Condition
Find the lateral surface area of a straight prism, at the base of which lies a rhombus with diagonals equal to 4\sqrt5 and 8, and a side edge equal to 5.
Solution
The area of the lateral surface of a straight prism is found by the formula S side. = P basic · h = 4a\cdot h, where P basic. and h, respectively, the perimeter of the base and the height of the prism, equal to 5, and a is the side of the rhombus. Let's find the side of the rhombus using the fact that the diagonals of the rhombus ABCD are mutually perpendicular and bisected by the point of intersection.
Schoolchildren who are preparing for passing the Unified State Exam in mathematics, you should definitely learn how to solve problems on finding the area of a straight and regular prism. Many years of practice confirm the fact that many students consider such geometry tasks to be quite difficult.
At the same time, high school students with any level of training should be able to find the area and volume of a regular and straight prism. Only in this case will they be able to count on receiving competitive scores based on the results of passing the Unified State Exam.
Key Points to Remember
- If the lateral edges of a prism are perpendicular to the base, it is called a straight line. All side faces of this figure are rectangles. The height of a straight prism coincides with its edge.
- A regular prism is one whose side edges are perpendicular to the base in which the regular polygon is located. Side faces of this figure are equal rectangles. A correct prism is always straight.
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