How to calculate length of a leg of a rectangular triangle

How to calculate length of a leg of a rectangular triangle

The triangle is called rectangular if the corner of one of its tops is equal 90 °. The party which lies opposite to this top is called a hypotenuse, and two others - legs. Lengths of the parties and size of corners in such figure are connected among themselves by the same ratios, as in any other triangle, but as the sine and a cosine of a right angle are equal to unit and zero, formulas considerably become simpler.


1. If lengths of one of legs (a) and a hypotenuse (c) a rectangular triangle are known, use Pythagorean theorem for calculation of length of the third party (b). Follows from it that required size has to be equal to a square root from the difference between squared by length of a hypotenuse and a square of length of the known leg: b = √ (with²-a²).

2. Knowing corner size (α) at the top of a triangle lying opposite to a leg of the known length (a) too it is possible to calculate the unknown length of the second leg (b). For this purpose apply definition of one of trigonometrical functions - a tangent - to an acute angle. From it follows that the required length of a leg has to be equal to the size of the known party divided into a tangent of an opposite corner: b = a/tg(α).

3. Use definition of a cotangent for an acute angle for finding of length of a leg (b) in case the corner size (β) adjoining other leg of the known length (a) is specified in conditions. The formula in a general view will look almost as well as in the previous step, replace in it only the name of function and designation of a corner: b = a/ctg(β).

4. With the known length of a hypotenuse (c) in calculations of the sizes of a leg (b) it is possible to use definitions of the main trigonometrical functions - a sine and a cosine - for acute angles. If in conditions the corner size (α) between these two parties is given, it is necessary to choose a cosine from two functions. Increase hypotenuse length by a cosine of the known corner: b = c*cos(α).

5. Use definition of a sine for acute angles when except length of a hypotenuse (c) the corner size (β) in the top lying opposite to a required leg (b) is given. The calculation formula in a general view will be similar to previous - it has to contain the work of length of a hypotenuse on a sine of the angle of the set size: b = c*sin(β).

Author: «MirrorInfo» Dream Team