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 need to figure out what kind it has.
General theory
A prism is any polyhedron whose sides are in the form of a parallelogram. Moreover, at its base can be any polyhedron - from a triangle to an n-gon. Moreover, the prism bases are always equal to each other. What does not apply to the side faces - they can vary significantly in size.
When solving problems, not only the prism base area is encountered. Knowledge of the lateral surface, i.e. all faces that are not bases, may be required. The full surface will already be the union of all the faces that make up the prism.
Sometimes in tasks appears height. She 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 direct prism or inclined does not depend on the angle between them and the side faces. If they have the same shapes in the upper and lower faces, then their areas will be equal.
Triangular prism
It has at the base a figure having three vertices, that is, a triangle. He, as you know, is different. If the triangle is rectangular, then it is enough to recall 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 base area of a triangular prism in general, formulas come in handy: Heron and one in which half of the side is taken to the height drawn to it.
The first formula should be written as follows: S = √ (p (p-a) (p-c) (p-c)). In this record, there is a half-perimeter (p), that is, the sum of three sides divided by two.
Second: S = ½ n a * a.
If you need to know the base area of the triangular prism, which is correct, then the triangle is equilateral. For it there is its own formula: S = ¼ a 2 * √3.
Quadrangular Prism
Its base is any of the known quadrangles. It can be a rectangle or square, box or rhombus. In each case, in order to calculate the base area of the prism, you will need your own formula.
If the base is a rectangle, then its area is defined as follows: S = av, where a, b are the sides of the rectangle.
When it comes to a quadrangular prism, the base area of the regular prism is calculated by the formula for the square. Because it is he who is the foundation. S = a 2 .
In the case when the base is a parallelepiped, such equality will be needed: S = a * n a . It happens that the side of the parallelepiped and one of the corners are given. Then, to calculate the height, you need to use the additional formula: n a = b * sin A. Moreover, the angle A is adjacent to the side "b", and the height n is opposite to this corner.
If a rhombus lies at the base of the prism, then to determine its area, the same formula will be needed as for a parallelogram (since it is its special case). But you can use this: S = ½ d 1 d 2 . Here d 1 and d 2 are two diagonals of a rhombus.
Regular pentagonal prism
This case involves breaking the polygon into triangles, whose areas are easier to find. Although it happens that the figures can be with 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
By the principle described for a pentagonal prism, it is possible to break 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 in it should the area of an equilateral triangle be multiplied by six.
The formula will look like this: S = 3/2 and 2 * √3.
Tasks
No. 1. Given a regular straight quadrangular prism. 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.
Decision. The base of the prism is a square, but its side is not known. You can find its value from the diagonal of the square (x), which is associated with the diagonal of the prism (d) and its height (n). x 2 = d 2 - n 2 . On the other hand, this segment “x” is the hypotenuse in the triangle, the legs of which 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 the double value of the base area and the quadruple lateral. The latter is easy to find by the formula for the 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 . The total surface area of the prism is 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 a regular triangular prism. At the base lies a triangle with a side of 6 cm. In this case, the diagonal of the side face is 10 cm. Calculate the area: base and side surface.
Decision. Since the prism is correct, its base is an equilateral triangle. Therefore, its area is 6 squared times 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 6 and 10 cm. To calculate their area, just multiply these numbers. Then multiply them by three, because there are so many lateral faces at the prism. Then the side surface area is a wound of 180 cm 2 .
Answer. Squares: bases - 9√3 cm 2 , lateral surface of the prism - 180 cm 2 .