Heights in a triangle call three pieces of straight lines, each of which is perpendicular to one of the parties and connects it to opposite top. At least two parties and two corners in an isosceles triangle have identical sizes therefore also lengths of two heights have to be equal. This circumstance considerably simplifies calculation of lengths of heights of a figure.
1. Hc which is carried out to the basis of an isosceles triangle can be calculated, knowing lengths of this basis (c) and side (a). For this purpose it is possible to use Pythagorean theorem as height, the side and a half of the basis form a rectangular triangle. Height and a half of the basis in it are legs therefore for the solution of a task take a root from the difference between squared by length of side and a quarter of a square of length of the basis: Hs = √ (a²-¼*c²).
2. The same height (Hc) can be calculated also on length of any of the parties if size at least of one corner is specified in conditions. If it is a corner at the triangle basis (α) and the known length determines the size of side (a), for receiving result multiply length of the known party and a sine of the known corner: Hs = a*sin(α). This formula follows from the theorem of sine.
3. If basis length is known (c) and the size of the corner adjoining to it (α), for calculation of height (Hc), increase a half of length of the basis by a sine of the known corner and divide into a difference sine between 90 ° and the size of the same corner: Hs = ½*c*sin(α)/sin (90 °-α).
4. At the known sizes of the basis (c) and opposite to it a corner (γ) for calculation of height (Hc) multiply a half of length of the known party by a difference sine between 90 ° and a half of the known corner, and you divide result into a sine of a half of the same corner: Hs = ½*c*sin (90 °-γ/2) / sin(γ/2). This formula, as well as two previous, follows from the theorem of sine in combination with the theorem of the sum of corners in a triangle.
5. Length of height which is carried out to one of sides (Ha) can be calculated, for example, knowing length of this party (a) and the area of an isosceles triangle (S). That to make it, find the doubled ratio size between the area and length of the known party: Ha = 2*S/a.