The pyramid is the polyhedron made of a certain number of the flat side surfaces and one basis having one general top. The basis, in turn, has one general edge with each side side and therefore its form determines the total number of sides of a figure. In the regular quadrangular pyramid of such sides five, but for calculation of full surface area it is enough to calculate the areas of only two of them.
Instruction
1. Full area surfaces of any polyhedron consists of the sum of the areas of its sides. In the regular quadrangular pyramid they are presented by two forms of polygons - in the basis the square lies, in side surfaces have a triangular configuration. Begin calculations, for example, with calculation of the area of the quadrangular basis of a pyramid (S ₒ). By definition of a regular pyramid in its basis the regular polygon, in this case - a square has to lie. If length of an edge of the basis (a) is specified in conditions, just build it in the second degree: S ₒ = a². If only length of diagonal of the basis (l) is known, for calculation of the area find a half of its square: S ₒ = l²/2.
2. Determine the area of a triangular side side of a pyramid of S ₐ. If length of its the edge (a), general with the basis, and an apothem (h) is known, calculate a half from the work of these two sizes: S ₐ = a*h/2. With lengths of a side edge (b) and edge of the basis (a) specified in conditions find a half of the work of length of the basis on a root from a difference between the squared length of a side edge and a quarter of a square of length of the basis: S ₐ = ½*a * √ (b²-a²/4). If except length of the edge (a), general with the basis, the flat corner in pyramid top is given (α), calculate the relation of the squared edge length to the doubled cosine of a half of a flat corner: S ₐ = a² / (2*cos(α/2)).
3. Having calculated the area of one side side (S ₐ), increase the received size four times to calculate the area of a side surface of the regular quadrangular pyramid. At the known apothem (h) and perimeter of the basis (P) it is action together with all previous step it is possible to replace with calculation of a half of the work of these two parameters: 4*S ₐ = ½*h*P. Anyway, put the received area of a side surface with the area of the square basis of a figure calculated on the first step is and there will be a full surface area of a pyramid: S = S ₒ+4*S ₐ.