How to find an unknown leg

How to find an unknown leg

The leg is the party of a rectangular triangle adjacent to a right angle. It is possible to find it, using Pythagorean theorem or the trigonometrical relations in a rectangular triangle. For this purpose it is necessary to know other parties or corners of this triangle.

It is required to you

  • - Pythagorean theorem;
  • - trigonometrical ratios in a rectangular triangle;
  • - calculator.


1. If in a rectangular triangle the hypotenuse and one of legs is known, then find the second leg, using Pythagorean theorem. As the sum of squares of legs of an and b, is equal to c hypotenuse square (with²=a²+b²), having made simple transformation, receive equality for finding of an unknown leg. Designate an unknown leg as b. To find it, find the difference of squares of a hypotenuse and the known leg, and from result allocate a root square b= √ (with²-a²).

2. Example. The hypotenuse of a rectangular triangle is equal to 5 cm, and one of legs of 3 cm. Find what the second leg is equal to. Substitute values in the removed formula and receive b= √ (5²-3²)= √ (25-9) = √ 16=4 cm.

3. If in a rectangular triangle length of a hypotenuse and one of acute angles is known, use properties of trigonometrical functions to find the necessary leg. If it is necessary to find the leg adjacent to the known corner to find it, use one of definitions of a cosine of the angle which says that it is equal to the relation of an adjacent leg of a to a hypotenuse of c (cos(α)=a/c). Then to find leg length, increase a hypotenuse by a cosine of a corner of a=c∙cos(α), adjacent to this leg.

4. Example. The hypotenuse of a rectangular triangle is equal to 6 cm, and its acute angle 30º. Find length a leg, adjacent to this corner. This leg will be equal to a=c∙cos(α) =6∙cos(30º) =6 ∙√ 3/2≈5.2 cm.

5. If the acute angle needs to find a leg opposite, use the same calculation procedure, only change a cosine of the angle in a formula for its sine (a=c∙sin(α)). For example, using a condition of the previous task, find length of a leg, opposite to an acute angle 30º. Having used the offered formula, receive: a=c∙sin(α) = 6∙sin(30º) = 6∙1/2=3 cm.

6. If one of legs and an acute angle is known, then for calculation of length of another use a tangent of angle which is equal to the relation of an opposite leg to adjacent. Then, if the leg of an is adjacent to an acute angle, find it, having divided an opposite leg of b into a tangent of angle of a=b/tg(α). If the leg of a protivolezhit to an acute angle, then it is equal to the work of the known leg of b on a tangent of an acute angle of a=b∙tg(α).

Author: «MirrorInfo» Dream Team