The pyramid is understood as one of types of polyhedrons which is formed of the polygon lying in the basis and triangles which are its sides and unite in one point - pyramid top. Will not force to find the area of a side surface of a pyramid special difficulty.
Instruction
1. First of all, it is worth understanding that the side surface of a pyramid is presented by several triangles which areas can be found by means of the most various formulas, depending on the known data: S = (a*h)/2 where h is height lowered on the party of a; S = a*b*sinβ where a, b are the parties of a triangle, and β - a corner between these parties; S = (r * (a + b + c))/2 where a, b, c are the parties of a triangle, and r is the radius of the circle entered in this triangle; S = (a*b*c)/4*R where R is the radius of the triangle circle described around; S = (a*b)/2 = r² + 2*r*R (if a triangle - rectangular); S = S = (a² * √ 3)/4 (if a triangle - equilateral). Actually, it is only the main of the known formulas for finding of the area of a triangle.
2. Having calculated by means of the formulas of the area of all triangles which are pyramid sides stated above it is possible to start calculation of the area of a side surface of this pyramid. It becomes extremely simply: it is necessary to put the areas of all triangles forming a side surface of a pyramid. A formula it can be expressed so: Sp = ΣSi where Sp is the area of a side surface of a pyramid, Si is the area i-oho of the triangle which is a part of its side surface.
3. For bigger clarity it is possible to review a small example: the regular pyramid which side sides are formed equilateral to triangles is given, and in its basis the square lies. Length of an edge of this pyramid is 17 cm. The area of a side surface of this pyramid is required to find. Decision: length of an edge of this pyramid is known, it is known that facet it - equilateral triangles. Thus, one may say, that all parties of all triangles of a side surface are equal to 17 cm. Therefore to calculate the area of any of these triangles, it will be required to apply a formula: S = (17² * √ 3)/4 = (289*1.732)/4 = 125.137 cm²Известно that in the basis of a pyramid the square lies. Thus, it is clear that these equilateral triangles four. Then the area of a side surface of a pyramid pays off so: 125.137 cm² * 4 = 500.548 cm²Ответ: the area of a side surface of a pyramid is 500.548 cm²
4. At first we will calculate the area of a side surface of a pyramid. The side surface is meant as the sum of the areas of all side sides. If you deal with a regular pyramid (that is such in which basis the regular polygon lies, and the top is projected in the center of this polygon), then for calculation of all side surface it is enough to increase basis perimeter (that is the sum of lengths of all parties of the polygon lying in the pyramid basis) by height of a side side (differently the called apothem) and to divide the received value into 2: Sb =1/2P*h where Sb is the area of a side surface, P - perimeter of the basis, h - height of a side side (apothem).
5. If before you any pyramid, then it is necessary to calculate separately the areas of all sides, and then to put them. As side sides of a pyramid are triangles, use a formula of the area of a triangle: S=1/2b*h where b is the triangle basis, and h is height. When the areas of all sides are calculated, it is necessary only to put them to receive the area of a side surface of a pyramid.
6. Then it is necessary to calculate the area of the basis of a pyramid. The choice of a formula for calculation depends on what polygon lies in the basis a pyramid: correct (that is such which all parties have identical length) or wrong. The area of a regular polygon can be calculated, having increased perimeter by the radius of the circle entered in a polygon and having divided the received value into 2: Sn=1/2P*r where Sn is the area of a polygon, P is a perimeter, and r is a radius of the circle entered in a polygon.
7. The truncated pyramid is a polyhedron which is formed by a pyramid and its section parallel to the basis. To find the area of a side surface of the truncated pyramid absolutely simply. Its formula is very simple: the area equals to the work of a half of the sum of perimeters of the bases on an apothem. Let's review an example of calculation of the area of a side surface of the truncated pyramid. The regular quadrangular pyramid is Let's say given. Lengths of the basis are equal to b=5 of cm, c = 3 cm. An apothem = 4 cm. To find the area of a side surface of a pyramid, it is necessary to find perimeter of the bases at first. It will be equal in the big basis to p1=4b=4*5=20 see. In the smaller basis the formula will be following: p2=4c=4*3=12 see. Therefore, the area will be equal: s=1/2 (20+12)*4=32/2*4=64 cm.
8. If in the basis of a pyramid the wrong polygon lies, for calculation of the area of all figure at first it will be necessary to break a polygon into triangles, to calculate the area of everyone, and then to put. In other cases to find a side surface of a pyramid, it is necessary to find the area of each its side side and to put the received results. In certain cases the problem of finding of a side surface of a pyramid can be facilitated. If one side side is perpendicular to the basis or two adjacent side sides are perpendicular to the basis, then the basis of a pyramid is considered an orthogonal projection of a part of its side surface, and they are connected by formulas.
9. To finish calculation of surface area of a pyramid, put the areas of a side surface and the basis of a pyramid.
10. The pyramid is a polyhedron, one of sides of which (basis) – any polygon, and other sides (side) – the triangles having the general top. On number of corners of the basis of a pyramid happen triangular (tetrahedron), quadrangular and so on.
11. The pyramid is the polyhedron having the basis in the form of a polygon, and other sides are triangles with the general top. An apothem is called height of a side distinction of a regular pyramid which is drawn from its top.