ABC Right Triangle Calculator
Solve a right triangle from two independent measurements: two sides, one side plus one acute angle, or the area plus one leg.
Known measurements
Choose the two measurements you already know.
All side lengths use one common unit; area uses its square.
Acute angles must be strictly between 0° and 90°.
Leg opposite Angle α. Enter a positive decimal using a dot.
Leg opposite Angle β. Enter a positive decimal using a dot.
Longest side, opposite the fixed 90° angle.
Acute angle opposite Side a.
Acute angle opposite Side b.
Use area with Side a or Side b to determine the other leg.
Solved triangle
How to use the ABC right triangle calculator
What this calculator does. This tool solves a labeled right triangle whose fixed right angle is opposite Side c (hypotenuse). It determines Side a, Side b, Side c (hypotenuse), Angle α, Angle β, and Area from two independent measurements. The governing relationships are the Pythagorean theorem, the fact that the two acute angles add to 90°, and the standard sine, cosine, and tangent ratios. It is intended for geometric calculation; it does not establish measurement accuracy, construction tolerances, or whether a real object is perfectly square.
When to use it. Typical uses include checking the diagonal of a rectangular layout from its rise and run, finding a missing leg from a measured hypotenuse, converting a known slope angle and one side into the remaining triangle dimensions, or recovering a second leg when a right-triangle area and one leg are known. For the underlying relationships, OpenStax gives a concise treatment of the Pythagorean theorem and right-triangle trigonometry.
How to calculate. The calculator opens with a complete 3 – 4 – 5 demonstration triangle, so the results and a validated example Excel workbook are available immediately.
- Use Given to choose the pair you know. The available combinations cover two sides, one side plus Angle α or Angle β, and Area plus either leg.
- Choose Length unit and Angle unit. All side measurements share the selected length unit, and Area uses the squared length unit. Changing a unit converts the current entered measurements rather than merely relabeling them.
- Replace the demonstration values in the two visible measurement fields. Results update live. Inputs accept positive decimal numbers with a dot; commas, scientific notation, unsupported symbols, zero, and negative values are rejected instead of silently reinterpreted.
- Read the complete solution in Solved triangle. Use Download Excel to export the current validated state as a real .xlsx workbook. Reset clears the demonstration and all measurement fields; after Reset, Excel export is disabled until a complete valid pair is entered again.
Input guide. Given is a required selection that defines which two measurements are independent. Choose the combination that matches what you actually measured; entering a hypotenuse smaller than a known leg is geometrically impossible. Length unit is required and applies consistently to Side a, Side b, and Side c (hypotenuse); for example, choose centimeters for a 3 cm by 4 cm classroom triangle. Angle unit is required and may be degrees or radians; changing it converts any active angle input. NIST notes the exact degree-to-radian relationship in its SI table of accepted units.
Side a is a required positive length whenever its selected Given combination uses it. It is the leg opposite Angle α. A realistic example is 3 cm. Increasing Side a while holding Side b fixed increases the hypotenuse, Area, and Angle α. Side b is the other leg, opposite Angle β; 4 cm is the startup example. Increasing Side b while holding Side a fixed increases the hypotenuse and Area but decreases Angle α. Side c (hypotenuse) is the longest side and must exceed a known leg; for example, 10 cm can be paired with a 6 cm leg. A common error is entering c smaller than a or b.
Angle α and Angle β are acute inputs when selected in Given. Each must be strictly between 0° and 90° (or the equivalent radian interval); for example, 30° is valid. Angle α is opposite Side a, while Angle β is opposite Side b. The two are complementary, so raising one forces the other lower. If you need a refresher on recovering an angle from a side ratio, OpenStax explains inverse trigonometric functions. Area is a required positive square-unit value only in the Area + side a or Area + side b modes. For example, 6 cm² with Side a = 3 cm gives Side b = 4 cm because Area = a × b ÷ 2. Area plus the hypotenuse is intentionally not offered because it can be ambiguous with respect to which leg receives which label.
Output guide. Side a, Side b, and Side c (hypotenuse) are solved lengths in the selected Length unit. Side c is always the longest valid side. Angle α and Angle β are the two acute angles in the selected Angle unit; they always sum to 90° for a valid solved model. Area is the exact geometric area implied by the solved legs and is displayed in the squared Length unit. These are mathematical consequences of the entered measurements, not statistical estimates. Extremely small or large values may be rejected to keep the calculation finite and exportable.
Worked example. The startup state selects Two sides: a and b, Length unit = centimeters, Angle unit = degrees, Side a = 3, and Side b = 4. The Pythagorean theorem gives c = √(3² + 4²) = 5 cm. Angle α = arctan(3 ÷ 4) = 36.870°, and Angle β = 90° – 36.870° = 53.130°. The area is 3 × 4 ÷ 2 = 6.000 cm². Those are the same values displayed on first open and written to the startup Excel workbook.
How the model works
For two known legs, the calculator uses c = √(a² + b²). If one leg and the hypotenuse are known, it rearranges the same identity, such as b = √(c² – a²). The hypotenuse condition is enforced before the square root is evaluated, so impossible pairs do not produce a misleading numeric result. MathWorld's right-triangle reference summarizes the same side identity and the fixed 90° angle.
For side-and-angle modes, the calculator applies SOH-CAH-TOA with labels fixed to this triangle: sin(α) = a/c, cos(α) = b/c, and tan(α) = a/b. The β relationships swap the two legs. Area modes use Area = a × b ÷ 2. Calculations are performed in canonical units at full floating-point precision; rounding is applied only when values are shown or formatted in the workbook.