Find the angles of a right triangle given any two sides. Compute both acute angles, all sides, area, perimeter, inradius, and trigonometric ratios.
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.
This right triangle angle calculator reduces manual rework when you need quick checks for assignments, exam prep, and design calculations. You can enter Known Sides, Decimal Places and immediately see dependent measurements, validity checks, and geometry relationships in one place. That makes it easier to catch input mistakes early and confirm your final answer before moving to the next step.
α = arctan(opposite / adjacent) β = 90° − α Hypotenuse: c = √(a² + b²) Area = ½ × opposite × adjacent Inradius = (a + b − c) / 2 Circumradius = c / 2 Height to hypotenuse = 2 × Area / c
Result: For pair=opp_adj, v1=3, v2=4, the tool returns the solved right triangle angle outputs shown in the result cards.
This example uses a realistic input set from the calculator workflow. After entry, the calculator applies the built-in right triangle angle formulas and reports derived values, checks, and classifications automatically.
This page is tailored to right triangle angle, with outputs tied directly to the form fields (Known Sides, Decimal Places). Instead of a one-line formula dump, it consolidates validation, derived metrics, and interpretation so you can solve and verify in one pass.
Use this tool for homework checks, worksheet generation, tutoring walkthroughs, and quick engineering geometry estimates. Presets and visual output blocks make it easier to compare scenarios and understand how each input affects the final result.
Keep units consistent, match each value to the correct field, and watch validity indicators before using the final numbers. If your case looks off, change one input at a time and use the output details to identify the mismatch quickly.
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).
Yes — the 45-45-90 isosceles right triangle has two equal legs. The hypotenuse is leg × √2.
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).
For a right triangle with legs a and b and hypotenuse c, the inradius is r = (a + b − c) / 2. Use this as a practical reminder before finalizing the result.
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.
The altitude from the right angle to the hypotenuse equals h = (leg₁ × leg₂) / hypotenuse = 2 × Area / c. Keep this note short and outcome-focused for reuse.