The area call a quantitative measure of the plane limited to perimeter of any two-dimensional figure. The surface of polyhedrons is made not less than of four sides, each of which can have own form and the sizes, so, and the area. Therefore calculation of total area of volume figures with flat sides not always a simple task.
Instruction
1. The full surface area of such polyhedrons as, for example, a prism, a parallelepiped or a pyramid consists of the sum of the areas of sides of different size and a form. These volume figures have side surfaces and the bases. Calculate separately the areas of these surfaces, proceeding from their form and the sizes, and then summarize the received values. For example, the total area (S) of six sides of a parallelepiped can be found doubling of the sum of works of length (a) on width (w), lengths on height (h) and width on height: S = 2 * (a*w + a*h + w*h).
2. The full surface area of a regular polyhedron (S) consists of the sum of the areas of each its side. As all side surfaces of this volume figure by definition have the identical forms and the sizes, it is enough to calculate the area of one side to have an opportunity to find total area. If from statements of the problem except number of side surfaces (N) length of any edge of a figure (a) and number of tops (n) of a polygon which forms each side is known to you, it is possible to make it with use of one of trigonometrical functions - a tangent. Find a tangent from the corner equal to the relation 360 ° to the doubled number of tops and increase result four times: 4*tg (360 (°/2*n)). Then at the received size divide the work of number of tops into a square of length of the party of a polygon: n*a² / (4*tg (360 (°/2*n))). It will also be the area of each side, and calculate the total area of a surface of a polyhedron, having increased it by number of side surfaces: S = N*n*a² / (4*tg (360 (°/2*n))).
3. In calculations of the second step-degree measures of corners are used, but often instead of them apply radian. Then it is necessary to make amendments proceeding from the fact that to a corner in 180 ° there corresponds the quantity a radian, equal to Pi's number to formulas. Replace a corner in 360 ° in formulas with the size equal to two such constants, and the total formula will even a little become simpler: S = N*n*a² / (4*tg (2*π / (2*n))) = N*n*a² / (4*tg(π/n)).