How to find the surface area of the pyramid?

What figure do we call a pyramid? First, it is a polyhedron. Secondly, an arbitrary polygon is located at the base of this polyhedron, and the sides of the pyramid (lateral faces) necessarily have the form of triangles converging at one common vertex. Now, figuring out the term, we'll figure out how to find the surface area of the pyramid.

It is clear that the surface area of such a geometric body will consist of the sum of the areas of the base and its entire lateral surface.

Calculation of the area of the base of the pyramid

The choice of the calculation formula depends on the form of the underlyingat the base of our polygon pyramid. It can be right, that is, with sides of the same length, or wrong. Let's consider both variants.

At the bottom is a regular polygon

From the school course we know:

the square of the square will be equal to the length of its side, squared;
the area of an equilateral triangle is equal to the square of its side divided by 4 and multiplied by the square root of the three.

But there is also a general formula for calculatingof the area of any regular polygon (Sn): multiply the perimeter of this polygon (P) by the radius of the circle (r) inscribed in it, and then divide the result into two: Sn = 1 / 2P * r.

At the bottom is an irregular polygon

The scheme of finding its area is,first to break the entire polygon into triangles, calculate the area of each of them by the formula: 1 / 2a * h (where a is the base of the triangle, h is the height that was lowered on this base), add all the results.

Surface area of the pyramid

Now calculate the area of the lateral surface of the pyramid, i.e. the sum of the areas of all its lateral sides. There are also 2 options.

Let us have an arbitrary pyramid, i.e. such, in the basis of which is an irregular polygon. Then we must calculate the area of each face separately and add the results. Since the sides of the pyramids by definition can only be triangles, the calculation follows the above formula: S = 1 / 2a * h.
Let our pyramid be the right one, i.e. in its base is a regular polygon, and the projection of the top of the pyramid is at its center. Then, to calculate the area of the lateral surface (Sb), it is sufficient to find half the product of the perimeter of the polygon-base (P) by the height (h) of the lateral side (the same for all faces): Sb = 1/2 P * h. The perimeter of a polygon is determined by adding the lengths of all its sides.

The total surface area of the regular pyramid is found by summing the area of its base with the area of the entire lateral surface.

Examples

For example, we calculate algebraically the surface areas of several pyramids.

The surface area of the triangular pyramid

At the base of such a pyramid is a triangle. By the formula So = 1 / 2a * h we find the area of the base. The same formula is used to find the area of each face of the pyramid, also having a triangular shape, and we obtain 3 areas: S1, S2 and S3. The area of the lateral surface of the pyramid is the sum of all areas: Sb = S1 + S2 + S3. Adding the areas of the sides and the base, we get the total surface area of the desired pyramid: Sn = S0 + Sb.

The surface area of the quadrilateral pyramid

The area of the lateral surface is the sum of 4-exterms: Sb = S1 + S2 + S3 + S4, each of which is calculated by the formula of the area of the triangle. And the base area will have to be searched, depending on the shape of the quadrilateral - right or wrong. The total surface area of the pyramid will again be obtained by adding the base area and the total surface area of the specified pyramid.