What Is a Right Triangle Calculator?
A right triangle calculator is a free online tool that solves any right triangle completely — finding all unknown sides, angles, area, perimeter, and altitude — from just two known values. It uses the Pythagorean theorem for sides and trigonometric functions (sin, cos, tan) for angle-side relationships, showing every step of the calculation.
Our right triangle calculator with angles accepts any combination of two values: two sides, one side and one angle, or two angles (though a right triangle always has one 90° angle, so knowing one other angle determines the third). The live diagram updates with your triangle's proportions after solving.
Solve right triangle calculator — Pythagorean theorem, sin/cos/tan, angles, area with complete step-by-step solutions
How to Solve the Right Triangle — All Input Combinations
Given: Two Legs (a and b)
Use the Pythagorean theorem to find the hypotenuse: c = √(a² + b²). Then use inverse trig to find the angles: A = arctan(a/b) and B = 90° − A.
📘 Example — legs a=3, b=4
c = √(3²+4²) = √(9+16) = √25 = 5
A = arctan(3/4) = arctan(0.75) = 36.87°
B = 90° − 36.87° = 53.13°
Area = ½ × 3 × 4 = 6
Given: One Leg + Hypotenuse (a and c)
Find the other leg: b = √(c² − a²). Find the angle: A = arcsin(a/c).
Given: One Leg + One Angle (a and A)
Use trig ratios: b = a / tan(A) and c = a / sin(A). Then B = 90° − A.
Remember: In a right triangle, the two non-right angles always add up to 90° (they are complementary). So if you know angle A, you automatically know B = 90° − A. This is why knowing just one acute angle + one side is enough to solve the entire triangle.
Is It a Right Triangle? FAQs
Is it a right triangle? / Is this a right triangle?
A triangle is a right triangle if and only if the square of its longest side equals the sum of squares of the other two sides — the Pythagorean theorem: c² = a² + b². Use our Check mode to test any three side lengths instantly.
How to tell if a triangle is a right triangle?
Follow these steps:
- Identify the longest side — this would be the hypotenuse (c) if it is a right triangle
- Calculate: a² + b² (sum of squares of the two shorter sides)
- Calculate: c² (square of the longest side)
- If a² + b² = c² → Yes, it is a right triangle
- If a² + b² ≠ c² → Not a right triangle
📘 Examples — Determine if a triangle is a right triangle
Sides 5, 12, 13: 5²+12² = 25+144 = 169 = 13² ✓ → RIGHT TRIANGLE
Sides 6, 8, 10: 6²+8² = 36+64 = 100 = 10² ✓ → RIGHT TRIANGLE
Sides 4, 5, 7: 4²+5² = 16+25 = 41 ≠ 49 = 7² ✗ → NOT a right triangle
Sides 7, 24, 25: 7²+24² = 49+576 = 625 = 25² ✓ → RIGHT TRIANGLE
Which side lengths form a right triangle?
Any three positive numbers satisfying a² + b² = c² form a right triangle. The most common Pythagorean triples (whole number right triangles) are:
- 3-4-5 and multiples: 6-8-10, 9-12-15, 12-16-20
- 5-12-13 and multiples: 10-24-26
- 8-15-17
- 7-24-25
- 20-21-29
- 9-40-41
Click any triple in our Check mode to instantly verify it forms a right triangle.
Right triangle uses — roof pitch, navigation bearings, construction angles, and physics force vectors
Real-Life Uses of Right Triangle Calculations
- Construction: Calculate roof pitch — if a roof rises 4 feet over a 12-foot run, the rafter length = √(4²+12²) = √160 ≈ 12.65 feet
- Navigation: If you travel 30 km east and 40 km north, your straight-line distance = √(30²+40²) = 50 km
- Screen size: A 1920×1080 pixel screen has a diagonal = √(1920²+1080²) ≈ 2202 pixels (≈ 27 inches at 82 PPI)
- Physics: Resolve force vectors using tan(angle) = vertical force / horizontal force
- Staircase design: Stair angle = arctan(rise/run) — a standard stair has rise=7", run=11", angle ≈ 32.5°
