Triangle and rectangle - two simplest flat geometrical figures in Euclidean geometry. In the perimeters formed by the parties of these polygons some site of the plane which area can be determined in many ways is concluded. The choice of a way in each case will depend on the known parameters of figures.
Instruction
1. Apply one of the formulas using trigonometrical functions to finding of the area of a triangle if sizes of one or several corners in a triangle are known. For example, at the known size of a corner (α) and to lengths of the parties, it making (In and C), the area (S) can be determined on a formula S= B*C*sin (α)/2. And at the known sizes of all corners (α, β and γ) and length of one party in addition (A) it is possible to use formula S= A²*sin (β)*sin(γ) / (2*sin(α)). If except all corners the radius (R) of a circumscribed circle is known, then use a formula S=2*R²*sin (α)*sin (β)*sin(γ).
2. If sizes of corners are not known, then for finding of the area of a triangle it is possible to use formulas without trigonometrical functions. For example, if height (H) which is carried out from the party which length is known (A) too is known use formula S=A*H/2. And if lengths of each of the parties are given (And, In and C), then at first find poluperimetr p= (And + B+C)/2, and then calculate the area of a triangle on a formula S= √ (p * (P-and) * (P-in) * (P-c)). If except lengths of the parties (And, In and C) the radius (R) of a circumscribed circle is known, then apply a formula S=A*B*C/(4*R).
3. For finding of the area of a rectangle it is possible to use trigonometrical functions too - for example if length of its diagonal (C) and the size of a corner which it makes from one of the parties (α) is known. In this case use a formula S=C²*sin (α)*cos(α). And if lengths of diagonals are known (C) and the size of a corner which they are (α) then apply a formula S=C²*sin (α)/2.
4. When finding the area of a rectangle it is possible to do without trigonometrical functions if lengths of its perpendicular parties are known (And yes C) - it is possible to apply a formula S=A*B. And if length of perimeter (P) and one party (A) is given, then use formula S= A*(P-2*A)/2.