The calculator finishes a right triangle. Angle C is 90°, side c is the hypotenuse, and the side opposite A is labeled a. You enter two independent parts. The result lists every remaining side and acute angle, with the Pythagorean step or the matching sine, cosine, or tangent ratio written out.
Known parts (C = 90°)
Lesson: Pythagoras and the sine, cosine, and tangent ratios
A right triangle has one 90° angle. The side opposite that angle is the hypotenuse, always the longest side. The Pythagorean theorem relates the three sides: the sum of the squares of the legs equals the square of the hypotenuse. That identity is enough when two sides are known.
When one acute angle is known, the three standard ratios finish the remaining sides. Sine equals opposite over hypotenuse. Cosine equals adjacent over hypotenuse. Tangent equals opposite over adjacent. The second acute angle is 90° minus the first, because the three interior angles still add to 180°.
This page does not treat an obtuse triangle, and it does not move the right angle away from C. If a homework problem places the right angle at A or B, relabel the vertices so that C is the right angle before you type the numbers.
Practice problems
1. Special triangle
Angle A is 30° and the opposite leg is 5. Find the remaining parts.
2. Two legs
The legs are 3 and 4. Find the hypotenuse and both acute angles.
3. Impossible leg
What does the calculator report if a = 10 and c = 6?
Show answers
Problem 1. B = 60°, c = 10, and b = 5√3, about 8.660254.
Problem 2. The hypotenuse is 5. Angle A opposite the side of length 3 is about 36.8699°, and B is about 53.1301°.
Problem 3. A leg cannot be longer than the hypotenuse. The calculator reports that error.