The area of the base of the prism: from triangular to polygonal

Different prisms are different from each other. At the same time, they have much in common. To find the area of the base of the prism, it will be necessary to understand what kind it has.

General theory

The prism is any polyhedron whose lateral sides have the form of a parallelogram. In this case, it can be any polyhedron - from a triangle to an n-gon. And the bases of the prism are always equal to each other. What does not apply to the side faces - they can vary significantly in size.

In solving problems, not only the area of the base of the prism is encountered. It may be necessary to know the lateral surface, that is, of all faces that are not bases. The complete surface will already be the union of all the faces that make up the prism.

Sometimes in tasks there is a height. It is perpendicular to the bases. The diagonal of a polyhedron is a segment that joins two vertices in pairs that do not belong to the same face.

It should be noted that the area of the base of the direct prism or oblique does not depend on the angle between them and the lateral faces. If they have the same figures in the upper and lower faces, then their areas will be equal.

Triangular prism

It has in the base a figure with three vertices, that is, a triangle. As you know, it happens to be different. If the triangle is rectangular, then it suffices to recall that its area is determined by half the product of the legs.

The mathematical notation is as follows: S = ½ av.

To find the area of the base of the triangular prism in general form, the following formulas will be useful: Heron and the one in which half 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 into two.

The second: S = ½ n _a * a.

If you want to know the area of the base of a triangular prism, which is correct, then the triangle is equilateral. For him, there is a formula: S = ¼ a ² * √3.

Quadrangular prism

Its basis is any of the known quadrangles. It can be a rectangle or a square, a parallelepiped or a rhombus. In each case, in order to calculate the area of the base of the prism, we need our own formula.

If the base is a rectangle, then its area is defined as: S = av, where a, and - sides of the rectangle.

When it comes to a quadrangular prism, the area of the base of the correct prism is calculated by the formula for the square. Because it is he who lies at the bottom. S = a ² .

In the case where the base is a parallelepiped, the following equality will be needed: S = a * n _a . It happens that the side of the parallelepiped is given and one of the corners. Then, to calculate the height, we need to use the additional formula: _a = b * sin A. Moreover, the angle A is adjacent to the "c" side, and the height is opposite to this angle.

If a rhombus lies at the base of the prism, then to determine its area, the same formula will be needed for the parallelogram (since it is its particular case). But we can also use this: S = 1 d ₁ d ₂ . Here d ₁ and d ₂ are the two diagonals of the rhombus.

Correct pentagonal prism

This case involves splitting the polygon into triangles whose areas are easier to learn. 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.

Correct hexagonal prism

According to the principle described for a pentagonal prism, it is possible to break the hexagon of the base into 6 equilateral triangles. The formula of the base area of such a prism is similar to the previous one. Only in it the area of an equilateral triangle should be multiplied by six.

The formula looks like this: S = 3/2 and ² * √3.

Tasks

№ 1. The correct straight quadrangular prism is given. 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. Find its value can be from the diagonal of the square (x), which is connected 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 - н ² ) / 2.

To replace d with 22, and "н" to replace it with 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 know the area of the entire surface, you need to add twice the value of the base area and quadruple side. The latter can be easily found from 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 . 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. The correct triangular prism is given. At the base lies a triangle with a side of 6 cm. At the same time, the diagonal of the lateral face is 10 cm. Calculate the areas: the base and the lateral surface.

Decision. Since the prism is correct, its base is an equilateral triangle. Therefore, its area is equal to 6 in the square 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 lateral faces are the same and represent rectangles with sides of 6 and 10 cm. To calculate their areas, it is enough to multiply these numbers. Then multiply them by three, because there are so many side edges of the prism. Then the area of the lateral surface is wound to 180 cm ² .

Answer. The area: the base is 9√3 cm ² , the lateral surface of the prism is 180 cm ² .