The area and perimeter - the main numerical characteristics of any geometrical figures. Finding of these sizes becomes simpler thanks to the standard formulas according to which it is also possible to calculate one through another with a minimum or total absence of additional initial data.
1. PryamougolnikZadach: find rectangle perimeter if it is known that the area is equal 18, and length of a rectangle is twice more than width. Decision: write down an area formula for a rectangle – S = a*b. On b statement of the problem = 2*a, from here 18 = a*2*a, a = √9 = 3. It is obvious that b = 6. On a formula the perimeter is equal to the sum of all parties of a rectangle – P = 2*a + 2*b = 2*3 + 2*6 = 6 + 12 = 18. In this task the perimeter coincided on value with the area of a figure.
2. KvadratZadach: find square perimeter if its area is equal to 9. Decision: on a formula of the area of a square of S = a^2, find length of the party of a = 3 from here. The perimeter is equal to the sum of lengths of all parties, therefore, of P = 4*a = 4*3 = 12.
3. TreugolnikZadach: any triangle of ABC which area is equal to 14 is given. Find triangle perimeter if B height which is carried out from top divides the triangle basis into pieces 3 and 4 cm long. Decision: on a formula the area of a triangle is a half of the work of the basis on height, i.e. S = ½*AC*BE. The perimeter is equal to the sum of lengths of all parties. Find length of the party of AC, having put lengths of AE and EC, AC = 3 + 4 = 7. Find BE triangle height = S*2/AC = 14*2/7 = 4. Consider a rectangular triangle of ABE. Knowing legs of AE and BE, it is possible to find a hypotenuse on Pythagoras's formula AB^2 = AE^2 + BE^2, AB = √ (3^2 + 4^2) = √25 = 5. Consider a rectangular triangle of BEC. On Pythagoras's formula BC^2 = BE^2 + EC^2, BC = √ (4^2 + 4^2) = 4 * √ 2. Now lengths of all parties of a triangle are known. Find perimeter from their sum P = AB + BC + AC = 5 + 4 * √ 2 + 7 = 12 + 4 * √ 2 = 4 * (3+ √ 2).
4. OkruzhnostZadach: it is known that the area of a circle is equal 16*π, find its perimeter. Decision: write down a formula of the area of a circle of S = π*r^2. Find r circle radius = √ (S/π) = √16 = 4. On a formula P perimeter = 2*π*r = 2*π*4 = 8*π. If to accept that π = 3.14, then P = 8*3.14 = 25.12.