The Right Triangle Calculator quickly finds the hypotenuse, area, perimeter, and angles α & β of any right triangle — whether you know two sides or one side and an angle.
It supports all major units (inches, centimeters, feet, meters) and automatically converts between them while showing accurate, copy-ready results.

Perfect for students, engineers, and builders, this calculator lets you apply the Pythagorean theorem and basic trigonometry instantly — no manual math required.

A right triangle is a type of triangle that has one 90° angle.
Its three sides are the two legs (a and b) — which form the right angle — and the hypotenuse (c), the longest side opposite the right angle.

This shape is the foundation of geometry, architecture, and trigonometry — used to calculate slopes, roof pitch, stair design, and distances in navigation or mapping.
For example, if a ladder reaches a 10 ft wall while its base is 6 ft away, the hypotenuse represents the ladder’s exact length.

Below are the key right triangle formulas you can use to find any missing side, angle, or area — essential for geometry, trigonometry, and construction projects.

• Pythagorean Theorem: c² = a² + b²
• Area: A = ½ab
• Perimeter: P = a + b + c
• Angles: α = arctan(a/b),
  β = arctan(b/a)
  (or using sin / cos relationships)
• Altitude to Hypotenuse: h = (a·b)/c

The Right Triangle Calculator includes four powerful modes — each designed to solve different scenarios depending on which sides or angles you know.
Choose the one that matches your given values to get instant, accurate results.

1. 1. Legs (a + b): When both legs are known.
   Calculates c = √(a² + b²),
   A = ½ab, and both acute angles α and β.
2. 2. Leg + Hypotenuse (c): When you know one leg and the hypotenuse.
   Finds the missing leg b = √(c² − a²),
   then computes area and angles automatically.
3. 3. Hypotenuse + Angle θ: Ideal for trigonometric setups.
   Uses a = c·cosθ,
   b = c·sinθ, and then calculates area, perimeter, and the second angle.
4. 4. Leg + Angle θ: When one leg and one angle are given.
   Solves using b = a·tanθ,
   c = a / cosθ, then derives all remaining sides and area.

Finding the area and hypotenuse of a right triangle is simple once you know any two sides.
Follow these steps or use the calculator to get precise results instantly.

• Area: Choose the two legs (base & height), then apply
  A = ½ab.
  The area represents the space enclosed by the triangle.
• Hypotenuse: Use the Pythagorean theorem
  c = √(a² + b²)
  or simply select the appropriate calculator mode (e.g., Legs Mode or Angle Mode) to compute it automatically.
• Angle-first Method: When one side and an angle are known, convert degrees ↔ radians as needed,
  then apply trigonometric formulas (sin, cos, tan) to find the remaining sides.

Triangles are classified by their sides and angles.
The table below highlights how each type differs — from the common right triangle to the perfectly balanced equilateral triangle.

Four quick scenarios that match the calculator’s modes. Values are rounded to two decimals where appropriate:

• Legs: a = 9, b = 12 →
  c = 15,
  A = 54,
  α = 36.87°,
  β = 53.13°
• Leg + c: a = 7, c = 25 →
  b = √(25² − 7²) = 24,
  A = ½ × 7 × 24 = 84
• c + θ: c = 10, θ = 30° →
  a = c·cosθ = 8.66,
  b = c·sinθ = 5
• a + θ: a = 5, θ = 40° →
  b = a·tanθ = 4.20,
  c = a / cosθ = 6.54

Use this quick conversion table to compare right triangle results in both inches and metric units.
The calculator converts between all systems automatically — this table helps visualize the relationships.

Keywords: area of right triangle in cm², triangle area converter.

The Right Triangle Calculator is not just for geometry class — it’s a practical tool for real-world applications where accuracy and quick calculations matter.

• Carpentry & Framing: Calculate cuts, rafter lengths, and support angles in woodworking or construction.
• Roof Pitch Design: Determine slope, rise, and run instantly — ideal for architects and roofing engineers.
• Layout & Surveying: Measure distances, plot land gradients, and ensure right-angled boundaries.
• CAD & Engineering: Integrate precise triangle data into technical drawings and 3D modeling software.
• Education & Training: Demonstrate trigonometry and Pythagorean relationships in classrooms or tutorials.

Even simple right triangle calculations can go wrong with small input errors.
Here are the most common mistakes to avoid when using the calculator:

• Mixing up the base and height — remember, the slanted side (hypotenuse) is never used as height.
• Forgetting that the right angle is always opposite the hypotenuse (the longest side in the triangle).
• Entering degree values into formulas expecting radians — always check your angle mode.
• Using a non-right triangle with this calculator — it’s specifically built for 90° triangles only.

1) What is a right triangle?
A triangle with one 90° angle. The two legs (a, b) form the right angle; the opposite side is the hypotenuse (c).

2) What is the longest side called?
The hypotenuse — it’s always opposite the right angle and is the triangle’s longest side.

3) How do you find the hypotenuse?
Use the Pythagorean theorem: c = √(a² + b²).
Or choose the calculator mode that matches your inputs (e.g., Legs or Angle mode).

4) What are the angles of a right triangle?
One angle is 90°; the other two are acute and satisfy α + β = 90°.
You can calculate them using trigonometric functions such as tan, sin, or cos.

5) How to find the area?
Use the formula A = ½ab, where a and b are the legs.
If you know one leg and an angle, find the other leg using trigonometry first.

6) What is the altitude of a triangle?
For a right triangle, the altitude to the hypotenuse is h = (a·b)/c.
It’s the perpendicular drawn from the right angle to the hypotenuse.

7) Is a right isosceles triangle possible?
Yes. It has angles of 45°–45°–90°, equal legs, and follows c = a√2.

8) Difference between scalene / isosceles / equilateral?
Scalene: All sides and angles are different (it can still be a right triangle).
Isosceles: Two equal sides and two equal angles (can also form a right triangle).
Equilateral: All sides and angles are equal (60° each; never a right triangle).

Explore more geometry calculators to continue learning and comparing shapes.
Each tool is designed for fast, accurate, and visual understanding of area, perimeter, and key dimensions.

• Triangle Area Calculator – for any 3-sided triangle.
• Rectangle Area Calculator – for precise 2D rectangles or squares.
• Circle Area Calculator – to find radius, diameter, and circumference.
• Trapezoid Area Calculator – for parallel-sided shapes.

Keywords: triangle calculators, pythagorean theorem reference.