Right Triangle Calculator: Solve for Sides, Angles, Area & Altitude
Free right triangle calculator: solve for all sides and angles from any two values. Pythagorean theorem, SOH CAH TOA trig ratios, area formula, altitude to hypotenuse. Special right triangles (30-60-90, 45-45-90). Step-by-step Logic Trace. Trusted by students and educators. All calculations local.
What This Right Triangle Calculator Does & Who It's For
Calculator Purpose & Ideal Users
- What You'll Get:All sides and angles, area (½ab), perimeter, and altitude to hypotenuse (h = ab/c). A dynamic SVG triangle that scales to your proportions. A Logic Trace with the exact Pythagorean and trig formulas used. Special right triangles (30-60-90, 45-45-90) are supported. No sign-up; calculations run locally.
- Ideal Users:Students: Geometry and trigonometry, homework help, step-by-step verification. Teachers: Demonstrate Pythagorean theorem and SOH CAH TOA with a live diagram. Construction & surveying: Diagonal lengths, slope angles, right-angle layout. Anyone: Quick triangle solver for sides and angles.
- Scope & Limits:Right triangles only. All sides positive. Hypotenuse must be longest. Angles α + β = 90°. Units: use any consistent linear measure (feet, meters); area in square units.
This right triangle calculator solves for all six attributes—sides a, b, c and angles α, β (γ = 90°)—from any two values, with at least one side. It combines the Pythagorean theorem and trigonometric ratios (SOH CAH TOA) in one right triangle solver. Built for accuracy and transparency: every step is shown in the Logic Trace.
Formulas: Pythagorean Theorem & SOH CAH TOA
The Pythagorean theorem:
a² + b² = c²
where c is the hypotenuse. For angles: sin(α) = a/c, cos(α) = b/c, tan(α) = a/b (SOH CAH TOA). Area: A = ½ab. Altitude to hypotenuse: h = ab/c. The Logic Trace shows the substitution for your specific inputs.How to Solve a Right Triangle: Input Combinations
Two sides: Given a and b → c = √(a² + b²). Given a and c → b = √(c² − a²). Given b and c → a = √(c² − b²). One side and one angle: Given a and α → c = a/sin(α), b = a/tan(α), β = 90° − α. Similar logic for (b,α), (a,β), (b,β), (c,α), (c,β). Enter any two valid values; the right triangle calculator selects the correct formula and displays the Logic Trace.
Special Right Triangles: 30-60-90 & 45-45-90
45-45-90 triangle: Both legs equal; hypotenuse = leg × √2. Ratio 1:1:√2. 30-60-90 triangle: Sides follow 1:√3:2. The calculator handles these and any right triangle—enter known sides or angles. Results may appear in simplified radical form when applicable (e.g. 5√2 for a 45° triangle).
Altitude to Hypotenuse & Real-World Use
The altitude from the right angle to the hypotenuse has length h = ab/c. It divides the triangle into two smaller similar right triangles. Common uses: find hypotenuse from two legs (e.g. ramp length), find angle from sides (e.g. roof pitch), diagonal of a rectangle, and surveying. Use consistent units (feet, meters); the tool does not convert.
Right Triangle Calculator FAQ
? What is a right triangle?
A right triangle has one 90° angle. The hypotenuse (c) is opposite that angle and is always the longest side. Legs a and b meet at the right angle; acute angles α and β sum to 90°. This free right triangle solver computes all six values from any two inputs (at least one side).
? How do I find the hypotenuse of a right triangle?
Given both legs: c = √(a² + b²). Example: a=3, b=4 → c=5. Or enter one leg and an angle—the tool uses SOH CAH TOA (sin, cos, tan) to find c. The Logic Trace shows the exact formula and substituted values for each solution.
? What is the altitude to the hypotenuse?
The altitude h from the right angle to the hypotenuse is h = ab/c. It splits the triangle into two smaller similar right triangles. The calculator displays it in the hero metrics with Area and Perimeter.
? Can I solve a right triangle with only angles?
Two angles alone don't set scale. Enter α and β (must sum to 90°) plus at least one side, or the tool assumes a=1 and computes b and c for demonstration. For real problems, always provide one side.
? Does this work for special right triangles (30-60-90, 45-45-90)?
Yes. Enter sides or angles for 30-60-90 (ratio 1:√3:2) or 45-45-90 (ratio 1:1:√2) triangles. The solver and dynamic diagram handle any right triangle. Results can appear in radical form where applicable.
? Is this right triangle calculator free and private?
Yes. Free to use, no sign-up. All calculations run locally in your browser—no data is sent or stored.