Right Triangle Angle Calculator — Find Angles from Two Sides

About the Right Triangle Angle Calculator — Find Angles from Two Sides

A right triangle has one 90° angle. Given any two of the three sides — opposite, adjacent, or hypotenuse — you can find both acute angles using inverse trigonometric functions and compute the missing side via the Pythagorean theorem. This is a cornerstone of practical trigonometry, used in surveying, navigation, construction, physics, and engineering.

If you know the opposite and adjacent sides, the angle is α = arctan(opposite/adjacent). If you know the hypotenuse and one leg, use arcsin or arccos. The complementary angle is always β = 90° − α. Once all sides and angles are known, you can derive the area (½ × base × height), perimeter, inradius (which equals (a + b − c)/2 for a right triangle), circumradius (which is always half the hypotenuse), and the altitude to the hypotenuse.

This calculator lets you select which pair of sides you know, computes both acute angles, fills in the missing side, and displays a full suite of properties. It also provides a trigonometric ratios table for both angles and a reference table of Pythagorean triples — integer-sided right triangles like 3-4-5, 5-12-13, and 8-15-17 that appear frequently in problems and real-world measurements.

When This Page Helps

Use this page when two side lengths are known and you want the full right-triangle solution. It keeps the inverse-trig angle calculation, missing side, and derived properties such as area, perimeter, inradius, and trig ratios together so the triangle can be checked as a whole.

How to Use the Inputs

1. Select which two sides you know: hypotenuse & opposite, hypotenuse & adjacent, or opposite & adjacent.
2. Enter the two side lengths.
3. The calculator finds the missing side and both acute angles.
4. View area, perimeter, inradius, circumradius, and height to hypotenuse.
5. Check the trig ratios table for sin, cos, tan of each angle.
6. Click a preset to load common right triangles.
7. Expand the Pythagorean triples reference for integer-sided examples.

Example Calculation

Tips & Best Practices

• The circumradius of any right triangle is exactly half the hypotenuse.
• For a 45-45-90 triangle, the legs are equal and the hypotenuse is leg × √2.
• For a 30-60-90 triangle, sides are in the ratio 1: √3: 2.
• If both legs are integers and the hypotenuse is also an integer, you have a Pythagorean triple.
• The altitude from the right angle to the hypotenuse equals (leg₁ × leg₂) / hypotenuse.

When To Use This Calculator

Find the angles of a right triangle given any two sides. Compute both acute angles, all sides, area, perimeter, inradius, and trigonometric ratios. Use it when you need a repeatable calculation in the math / geometry category and want the setup, result, and supporting values kept together. This is especially helpful when small input changes, unit choices, or rounding decisions can change the final number.

How To Check The Result

Start by confirming that the inputs match the formula shown on the page. Then compare the main output with the worked example and any secondary values shown by the calculator. If the result will be used in another calculation, keep extra precision until the final step and record the assumptions beside the number.

Practical Notes

Treat the result as a calculation aid rather than a substitute for context. For schoolwork, include the formula and substitution steps. For planning, technical, financial, or health-related decisions, verify important numbers against primary records, current rules, or a qualified professional before acting on them.

Frequently Asked Questions

• How do I find the angle if I know two sides?

  Use inverse trig functions. If you know opposite and adjacent: α = arctan(opp/adj). If you know opposite and hypotenuse: α = arcsin(opp/hyp). If adjacent and hypotenuse: α = arccos(adj/hyp).

• Can a right triangle have two equal sides?

  Yes — the 45-45-90 isosceles right triangle has two equal legs. The hypotenuse is leg × √2.

• What is a Pythagorean triple?

  A set of three positive integers (a, b, c) where a² + b² = c². The smallest is (3, 4, 5). Others include (5, 12, 13), (8, 15, 17), and (7, 24, 25).

• What is the inradius of a right triangle?

  For a right triangle with legs a and b and hypotenuse c, the inradius is r = (a + b − c) / 2.

• Why is the circumradius half the hypotenuse?

  By Thales' theorem, any triangle inscribed in a semicircle with the diameter as one side is a right triangle. So the hypotenuse is the diameter, and R = c/2.

• How do I find the height to the hypotenuse?

  The altitude from the right angle to the hypotenuse equals h = (leg₁ × leg₂) / hypotenuse = 2 × Area / c.