Such triangle which two parties are equal among themselves is called isosceles. All formulas intended for determination of the area of any triangle are fair as well for isosceles. However formulas of the area of an isosceles triangle have simpler appearance and are sometimes more convenient in calculations.
It is required to you
- trigonometrical ratios
Instruction
1. By height of an isosceles triangle usually mean length of the perpendicular lowered on the "unequal" party, and by the basis – length of this party. For finding of the area of an isosceles triangle designate length of its equal parties through and, basis length – through with, and height length – in century. In this case, the formula for calculation of the area (P) will look as follows: П = ½ * with * in
2. To find a formula of the area of an isosceles triangle through the basis and length of the equal party, use Pythagorean theorem and the fact that the basis is halved by height. The following expression for height turns out: in = √ (and² - with²/4), having substituted it in the above-stated formula, receive: П = ½ * with * √ (and² - with²/4).
3. For finding of the area of an isosceles triangle on the basis of Heron's formula substitute in it lengths of the parties of an isosceles triangle taking into account that two of them are equal. After a number of reductions it will turn out: П = ½ * with * √ [(and - ½с) * (and + ½с)]. It is easy to notice that both formulas are identical as the difference of squares in the first formula just decayed on the work of the sum and a difference.
4. To find a formula of the area of an isosceles triangle through values a catch, designate: α - a corner between the equal parties and the basis; γ-a corner between equal sides. Then, using elementary trigonometrical ratios, receive: П = ½ * and * with * cos(γ/2), P = ½ * with * and * sin(α/2), P = ½ * with²/tg(γ/2), P = ½ * with² * tg(α/2), P = and² * sin(γ/2) * cos(γ/2), P = and² * sin(α/2) * cos(α/2),
5. The above-stated formulas cover all main options of calculation of the area of an isosceles triangle. However if to consider that height of an isosceles triangle is at the same time its bisector and a median, then it is possible "to remove" couple more of formulas, having replaced vp = ½ * with * voboznacheny heights with designation of a median or bisector.