Right Triangle Side and Angle Calculator
Solve right triangles instantly by calculating missing sides, angles, hypotenuse, area, and perimeter using trigonometry and the Pythagorean theorem.
Calculate NowTwo Legs Known
Enter the two legs (a and b) of the right triangle.
Hypotenuse + Angle
Enter the hypotenuse and one acute angle (degrees).
Side + Angle
Area + Side
Enter the area and one leg; we'll find the other leg.
Triangle Visualization
Adjust the legs and see your triangle rendered to scale.
Right Triangle Formulas
The essential equations every right triangle solver relies on.
Pythagorean Theorem
a² + b² = c²Relates the two legs (a, b) to the hypotenuse (c). The square of the hypotenuse equals the sum of the squares of the legs.
Sine
sin θ = Opposite ÷ HypotenuseUse sine when you know an angle and the hypotenuse, or to find an angle from the opposite side and hypotenuse.
Cosine
cos θ = Adjacent ÷ HypotenuseCosine links an angle to its adjacent leg and hypotenuse — perfect for resolving horizontal components.
Tangent
tan θ = Opposite ÷ AdjacentTangent compares the two legs directly. Ideal when both legs are known or for slope calculations.
Area
Area = (a × b) ÷ 2Right triangles make area easy: the two legs serve as the base and height, so just multiply and halve.
Perimeter
P = a + b + cAdd all three sides. If the hypotenuse is unknown, calculate it with the Pythagorean theorem first.
What Is a Right Triangle?
A foundational shape in geometry, trigonometry, and engineering.
The Right Angle
A right triangle contains exactly one 90° angle. The other two angles are acute and always sum to 90°, since every triangle's interior angles total 180°.
Legs
The two sides forming the right angle are called legs (often labeled a and b). They serve as the base and height when calculating area.
Hypotenuse
The side opposite the right angle is the hypotenuse (c). It is always the longest side and is found with the Pythagorean theorem.
Acute Angles
The two non-right angles (α and β) are acute. Knowing one immediately gives the other: β = 90° − α.
Special Triangles
The 3-4-5, 5-12-13, and 8-15-17 are common Pythagorean triples — sets of integers that satisfy a² + b² = c² exactly.
Why It Matters
Right triangles model ramps, roofs, screens, ladders, and countless physics problems. They underpin all of trigonometry.
How to Solve a Right Triangle
Choose the method that matches the values you already know.
Two Sides Known
If you know both legs, apply the Pythagorean theorem: c = √(a² + b²). Then find the angles using α = arctan(a/b) and β = 90° − α. Area is (a·b)/2.
One Side + One Angle
Pick the trig ratio that links your known side to the angle: sine for opposite, cosine for adjacent, tangent when both legs are involved. Rearrange to isolate the unknown side.
Hypotenuse + Angle
Use a = c·sin(α) and b = c·cos(α). Then β = 90° − α. This is the fastest case because both legs come from one multiplication.
Area + One Side
From Area = (a·b)/2, isolate the missing leg: b = 2·Area / a. Once both legs are known, fall back to the Pythagorean theorem for the hypotenuse.
The Pythagorean Theorem Explained
Geometry's most famous equation, demystified.
Named after the Greek mathematician Pythagoras (c. 570–495 BCE), this theorem states that in any right triangle the square of the hypotenuse equals the sum of the squares of the other two sides: a² + b² = c².
Geometrically, if you build a square on each side of a right triangle, the area of the largest square (on the hypotenuse) is exactly equal to the combined area of the two smaller squares. This visual proof has been demonstrated in hundreds of different ways across cultures and centuries.
The theorem is essential whenever you need to find a missing length — in surveying, construction, navigation, and even computer graphics where 2D and 3D distances are computed millions of times per second.
Worked Example
Trigonometry & Right Triangles (SOHCAHTOA)
The mnemonic that unlocks every trig problem.
Sine = Opposite / Hypotenuse
For an acute angle θ, divide the length of the side opposite that angle by the length of the hypotenuse.
Cosine = Adjacent / Hypotenuse
Cosine uses the leg next to the angle (but not the hypotenuse) divided by the hypotenuse.
Tangent = Opposite / Adjacent
Tangent skips the hypotenuse entirely — just opposite over adjacent. Great for slope and pitch.
To recover an angle from two known sides, use the inverse functions: arcsin, arccos, or arctan. Always confirm whether your calculator is in degree or radian mode before computing.
Worked Examples
Step-by-step solutions for the most common right triangle scenarios.
Example 1 — Classic 3-4-5 Triangle
α = arctan(3/4) ≈ 36.87° → β = 90° − 36.87° = 53.13°
Area = (3 × 4) / 2 = 6 | Perimeter = 3 + 4 + 5 = 12
Example 2 — Hypotenuse and Angle
b = 10·cos(30°) = 10·0.8660 ≈ 8.660
β = 90° − 30° = 60°
Area = (5 × 8.660) / 2 ≈ 21.65 | Perimeter ≈ 23.66
Example 3 — Area and One Side
c = √(6² + 8²) = √100 = 10
α = arctan(6/8) ≈ 36.87° → β ≈ 53.13°
Perimeter = 6 + 8 + 10 = 24
Real-World Applications
Right triangles are everywhere — far beyond the math classroom.
Construction
Framers use the 3-4-5 rule to confirm square corners; roofers calculate rafter lengths from rise and run.
Architecture
Architects compute stair stringers, ramps, beam diagonals, and ADA-compliant slope angles.
Surveying
Total stations and theodolites resolve distances and elevations through right triangle trigonometry.
Engineering
Vector decomposition, force diagrams, and stress analysis rely on resolving values along perpendicular axes.
Navigation
Pilots and sailors compute bearings, ground speed, and triangulated positions with sine and cosine.
Physics
Projectile motion, inclined planes, and electromagnetic wave components all split into right-triangle parts.
Roof Pitch
Roof pitch is simply rise/run = tan(θ). Convert any pitch into degrees in one tan⁻¹ step.
Land Measurement
Farmers and surveyors break irregular plots into right triangles to compute precise areas.
Why Use a Right Triangle Calculator?
Instant Calculations
Solve in milliseconds what would take minutes by hand — and avoid square-root and trig keystroke errors.
Eliminate Mistakes
Built-in formulas remove the risk of misremembering SOHCAHTOA or flipping a ratio.
Student-Friendly
See the formula used and a step-by-step breakdown — perfect for checking homework.
Engineering Accuracy
Results to four decimal places are precise enough for drafting, construction, and physics labs.
Mobile Access
Fully responsive — use it on a phone at a job site, in a workshop, or at your desk.
Time Savings
Bulk-check answers, prototype designs, or estimate quotes without ever opening a separate scientific calculator.
Frequently Asked Questions
How do I calculate a right triangle?
Identify what you know — two sides, a side and an angle, or area and a side. Then apply the matching formula: Pythagorean theorem for sides, sine/cosine/tangent for side-angle pairs, or rearranged area for a missing leg. Our calculator handles all cases automatically.
What is the hypotenuse?
The hypotenuse is the side opposite the right (90°) angle. It is always the longest side of a right triangle and is computed using c = √(a² + b²).
How do I use the Pythagorean theorem?
Square each leg, add them together, and take the square root: c = √(a² + b²). To find a missing leg instead, rearrange: a = √(c² − b²).
What is SOHCAHTOA?
It's a mnemonic for the three primary trig ratios: Sine = Opposite/Hypotenuse, Cosine = Adjacent/Hypotenuse, Tangent = Opposite/Adjacent.
Can I calculate angles from side lengths?
Yes. Use the inverse trig functions: α = arctan(opposite/adjacent), or arcsin(opposite/hypotenuse), or arccos(adjacent/hypotenuse). The remaining acute angle is simply 90° minus the one you just found.
How do I find area and perimeter?
For a right triangle, Area = (a × b) / 2 because the two legs serve as base and height. Perimeter is just a + b + c — the sum of all three sides.
What is the difference between degrees and radians?
Both measure angles. Degrees divide a full rotation into 360 parts; radians use 2π. To convert, multiply degrees by π/180 to get radians, or radians by 180/π for degrees.
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Solve Any Right Triangle Instantly
Use our free Right Triangle Side and Angle Calculator to calculate missing dimensions, angles, area, and perimeter with confidence.
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