Triangle – one of the most widespread and studied geometrical figures. For this reason there is a set of theorems and formulas on finding of its numerical characteristics. To find the area of any triangle if three parties are known, it is possible on Heron's formula.
Instruction
1. Heron's formula – the real find at the solution of mathematical tasks, it helps to find the area of any any triangle (except degenerated) if its parties are known. This Ancient Greek mathematician was interested in a triangular figure only with integer measurements which area is also an integer, however it does not prevent today's scientists and also school students and students to apply it to any other.
2. To use a formula, it is necessary to know one more numerical characteristic – perimeter, to be exact, poluperimetr a triangle. It is equal to the half-sum of lengths of all its parties. It is required to simplify a little the expression which is quite bulky: S = 1/4 · √ ((AV + VS + AC) • (VS + AC are AV) • (AV + AC are VS) • (AV + VS are AC)) r = (AB+BC+AC)/2 – poluperimetr; S = √ (r • (р - AV) • (р - VS) • (р - AC)).
3. Equality of all parties of a triangle which in this case is called correct turns a formula into simple expression: S = 3 · and²/4.
4. The isosceles triangle is characterized with an identical length of two of three parties of AV = VS and, respectively, adjacent corners. Then Heron's formula will be transformed to the following expression: S = 1/2 · AC • √ ((AV + 1/2•AC) • (AC – 1/2·AB)) = 1/2•AC • √ (AV² – 1/4•AC²), where AC – length of the third party.
5. It is possible to determine the area of a triangle by three parties not only by means of Heron. For example, let r radius circle is entered in a triangle. It means that it concerns all its parties which lengths are known. Then the area of a triangle can be found on the formula too connected with poluperimetry and consists in his simple work on the radius of the entered circle: S = 1/2 · (AV + VS + AC) = p·r.
6. Example on application of a formula of Heron: let the triangle with the parties and =5 be set; b=7 and with =10. Find the area.
7. ReshenieVychislite poluperimetr: р = (5 + 7 + 10) = 11.
8. Calculate required size: S = √ (11 • (11-5) • (11-7) • (11-10)) ≈ 16.2.