Square call the flat geometrical figure made of four parties of identical length which form tops with the corners equal 90 °. It is a regular polygon, and calculation of parameters of such figures is much simpler, than similar figures with any sizes of corners in tops. In particular, calculation of the surface area limited to the parties of a square can be made by a large number of ways on very simple formulas.

## Instruction

1. The simplest the formula of calculation of the area of a square (S) will be in case length of the party (a) of this figure is known - just increase it by yourself (square): S = a².

2. If in statements of the problem length of perimeter (P) of this figure is given, it is necessary to add one more mathematical operation to the formula given above. As the perimeter consists of the sum of lengths of all parties of a polygon, it contains in a square four identical composed, i.e. length of each party can be written down as P/4. Substitute this value in a formula of the previous step. At you such equality has to turn out: S = P²/4² = P²/16.

3. Diagonal of a square (L) connects two of its opposite tops, forming together with two parties rectangularof triangle. This property of a figure allows with use of Pythagorean theorem (L²=a²+a²) on length of diagonal to calculate length of the party (a=L / √ 2). Substitute also this expression in the same formula from the first step. In a general view the decision has to look so: S = (L/√ 2)² = L²/2.

4. It is possible to calculate the area of a square and on diameter (D) of the circle described about it. As the diagonal of any regular polygon coincides with diameter of a circumscribed circle, in a formula of the previous step replace only designation of diagonal with designation of diameter: S = D²/2. If it is necessary to express the area not through diameter, and through radius (R), transform equality thus: S = (2*R)²/2 = 2*R².

5. Calculation of the area on diameter (d) of an inscribed circle the little is more difficult as in relation to a square this size is always equal to length of its party. As well as in the previous step, for receiving a formula of calculations you need to replace only designation in the equality which is already described above - this time involve identity from the first step: S = d². If necessary to use radius (r) instead of diameter, transform this formula so: S = (2*r)² = 4*r².