How to Find the Legs and Hypotenuse of a Right Triangle with One Leg and Angles

Knowing one leg and all the angles in a right triangle can help determine the lengths of the other leg and the hypotenuse using trigonometric functions. This guide will walk you through the process step-by-step, ensuring accuracy and understanding.

Understanding the Triangle

A right triangle consists of two legs and a hypotenuse. The hypotenuse is the longest side, opposite the right angle. The two angles adjacent to the right angle add up to 90 degrees. Label the right angle as C, the angle opposite the known leg as A, and the remaining angle as B.

Step-by-Step Calculation

Step 1: Identify the Angles

Recognize that C 90°, as it is the right angle.

Use the triangle angle sum property: A B 90°

Step 2: Use Trigonometric Ratios

For the unknown leg b adjacent to angle A, use the tangent function:

b  a / tan(A)

For the hypotenuse c, use the sine function:

c  a / sin(A)

Step 3: Calculate the Lengths

Substitute the known values of a and A into the formulas to find b and c.

Example Calculation

Suppose you know:

One leg: a 5

Angle A: A 30°

To find leg b:

b  5 / tan(30°)  5 / (1 / sqrt(3))  5sqrt(3) ≈ 8.66

To find the hypotenuse c:

c  5 / sin(30°)  5 / 0.5  10

Summary

With one leg and the angles, you can easily find the other leg and the hypotenuse using tangent and sine functions. Make sure you correctly identify which leg corresponds to which angle.

Other Triangle Types and Basic Trigonometry

For specific types of right triangles like 45:45:90 or 30:60:90, the ratio of the sides is straightforward:

45:45:90 Triangle: The ratio of the sides is 1:1:√2

30:60:90 Triangle: The ratio of the sides is 1:√3:2

Using basic trigonometric ratios (SOH CAH TOA):

Sine (SOH): sin(A) opposite / hypotenuse

Cosine (CAH): cos(A) adjacent / hypotenuse

Tangent (TOA): tan(A) opposite / adjacent

For example, if you know a known angle of 30° and the side length opposite it is 10 cm, you can use the sine function to find the hypotenuse:

Sin(30°)  0.5  10 / HH  10 / 0.5  20 cm

Using a ruler, you can also measure the third side of the triangle to confirm the lengths using these trigonometric ratios.

The hypotenuse is always the longest side in a right angle triangle, opposite the 90-degree angle.