Right Triangle Sides & Angles Calculator

Enter one side and one acute angle to find all other sides and the other angle using trigonometry, with step-by-step solutions.

Enter One Side & One Angle

Angle A is opposite leg a. Angle B is opposite leg b. The right angle C = 90 degrees is opposite the hypotenuse c.

Results

Area
--
square units
Leg a--
Leg b--
Hypotenuse c--
Angle A--
Angle B--
Angle C90 degrees
Perimeter--

Step-by-Step Solution

Solving Right Triangles with Trigonometry

When you know one side and one acute angle of a right triangle, you can determine all other sides and angles using trigonometric functions. Since one angle is always 90 degrees, the other acute angle is simply 90 minus the known angle. The three sides are then found using sine, cosine, and tangent.

Trigonometric Relationships

Sine (sin)

Ratio of the opposite side to the hypotenuse.

sin(A) = opposite / hypotenuse = a / c

Cosine (cos)

Ratio of the adjacent side to the hypotenuse.

cos(A) = adjacent / hypotenuse = b / c

Tangent (tan)

Ratio of the opposite side to the adjacent side.

tan(A) = opposite / adjacent = a / b

How to Use This Calculator

  1. Select which side you know (leg a, leg b, or hypotenuse c) and enter its length.
  2. Enter the known acute angle in degrees (must be between 0 and 90).
  3. Specify which angle it is (angle A opposite leg a, or angle B opposite leg b).
  4. Click Calculate to find all unknown sides and the other acute angle.

Solving When Leg a Is Known with Angle A

If you know leg a and angle A: the other angle B = 90 - A. Then leg b = a / tan(A) and hypotenuse c = a / sin(A). This uses the tangent and sine functions respectively.

Solving When Hypotenuse Is Known with Angle A

If you know hypotenuse c and angle A: leg a = c x sin(A) and leg b = c x cos(A). The other angle B = 90 - A. This is perhaps the most straightforward case since sine and cosine directly give the legs.

Common Angle Values

  • 30 degrees: sin = 0.5, cos = 0.866, tan = 0.577
  • 45 degrees: sin = 0.707, cos = 0.707, tan = 1.0
  • 60 degrees: sin = 0.866, cos = 0.5, tan = 1.732

Applications

This type of calculation is essential in navigation (bearing and distance problems), construction (determining roof pitches and ramp lengths), physics (resolving force vectors), and surveying (measuring heights and distances using angles of elevation or depression).