Two Sides
Enter a and b, or a and c, or b and c. The third side is calculated via a² + b² = c² or its rearrangements.
Algebra & equation solver
Right Triangle Lab: Pythagorean & Trig Solver
Solve right triangles from any two values (sides or angles). Pythagorean theorem, trig ratios, area, perimeter, altitude. Dynamic SVG visualizer.
By Jeff Beem
Updated
Right Triangle Lab
Enter any two values (at least one side). Sides a, b, and hypotenuse c; angles α, β (γ = 90°). Calculated fields use a dashed stone border.
Triangle results
Area
6
A = ½ab
Perimeter
12
P = a + b + c
Altitude h
2.400000
h = ab/c
a3
b4
c5
α36.8699°
β53.1301°
Right triangle
Logic trace
- Pythagorean Theorema² + b² = c² ⇒ 3.00² + 4.00² = 9.00 + 16.00 = 25.00
- Solve for cc = √(a² + b²) = √25.00 ≈ 5.00
- AreaA = ½ab = ½ × 3.00 × 4.00 = 6
- Trig ratiossin(α) = a/c = 3.00/5.00, cos(α) = b/c, tan(α) = a/b
- Altitude to hypotenuseh = ab/c = (3.00 × 4.00) / 5.00 = 2.40
How to Use the Right Triangle Lab
Enter any two values (at least one side). The solver uses the Pythagorean theorem and trig ratios automatically. The dynamic diagram scales to your inputs; the Logic Trace explains each step. Trusted by students and educators.
Quick Tips
Side + Angle
Enter one side and α or β. SOH CAH TOA gives the rest (e.g. c = a/sin(α), b = a/tan(α)).
Hero Metrics
Area (½ab), Perimeter, and Altitude (h = ab/c) appear in the dashboard. Calculated fields use a dashed mint border.
Special Triangles
Works for 30-60-90 and 45-45-90. Enter known sides or angles, the tool solves for the rest.
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
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.
- 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.
How to Use This Calculator
Enter any two of the five fields: Side a, Side b, Hypotenuse c, Angle α, or Angle β. At least one value must be a side—two angles alone do not set the triangle's scale. The solver automatically selects the Pythagorean theorem or the appropriate trig ratio (SOH CAH TOA) depending on your inputs, and all six attributes update at once.
- Two Sides Known:Enter any pair of sides (a and b, a and c, or b and c). The third side is computed via a² + b² = c² or its rearrangement, and both acute angles are derived from inverse trig functions.
- One Side + One Angle:Enter a side and either α or β. The solver uses sin, cos, or tan to find the remaining sides and the complementary angle (α + β = 90°).
- Hero Metrics:The dashboard displays Area (½ab), Perimeter (a + b + c), and Altitude to the hypotenuse (h = ab/c). Calculated fields are marked with a dashed mint border.
- Diagram and Logic Trace:The dynamic SVG triangle scales to your proportions with labeled sides and angles. The Logic Trace shows the exact formula used and every substituted value, so you can follow the solution step by step.
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.
Mathematical Reference Note
Calculation Logic: This tool uses standard mathematical algorithms. While we strive for accuracy, errors in logic or user input can result in incorrect data.
Verification: Results should be cross-checked if used for important academic, professional, or personal calculations.
Standard Terms: This tool is provided free of charge and as-is. CalcRegistry provides no warranty regarding the accuracy or fitness of these results for your specific needs.
© 2026 CalcRegistry Reference Last System Check: May 2026Free Online Utility Tools