Angle of Right Triangle Calculator

Enter the two legs of a right triangle to find both acute angles, the hypotenuse, and the area using the tangent ratio and the Pythagorean theorem.

Quick Facts

Angle formula
tan(A) = a / b, tan(B) = b / a
The right angle at C is always 90°, so A and B always sum to 90°.
Hypotenuse formula
c = √(a² + b²)
The Pythagorean theorem, valid only because C is a right angle.

Your Results

Calculated
Angle A
-
Opposite leg a: arctan(a/b)
Angle B
-
Opposite leg b: arctan(b/a)
Hypotenuse (c)
-
√(a² + b²)
Area
-
½ × a × b

Ready

Enter both legs of the right triangle, then press Calculate.

Formula and method for the angle of a right triangle

A right triangle has one 90° angle (at vertex C) and two acute angles (A and B) that always add up to the remaining 90°. If you know the lengths of the two legs — the two sides that form the right angle — you can find both acute angles with the tangent ratio, plus the hypotenuse with the Pythagorean theorem. No protractor is required.

How the calculation works

Label the leg opposite angle A as a and the leg opposite angle B as b. Because the two legs are perpendicular, the tangent of each acute angle is simply the ratio of the opposite leg to the adjacent leg: tan(A) = a / b and tan(B) = b / a. Taking the inverse tangent (arctan) of each ratio gives the angle itself: A = arctan(a/b), B = arctan(b/a). The hypotenuse follows from the Pythagorean theorem, c = √(a² + b²), and the area of any right triangle is half the product of its legs, Area = ½ × a × b, since one leg can serve as the base and the other as the height.

Common mistakes

  • Mixing units: enter both legs in the same unit (both in centimeters, or both in inches). Mixing units before calculating gives a meaningless ratio and a wrong angle.
  • Entering the hypotenuse as a leg: this calculator expects the two legs that meet at the right angle, not the hypotenuse. The hypotenuse is always the longest side and is opposite the 90° angle — never enter it into either leg field.
  • Mixing up degrees and radians: arctan in most programming languages and calculators returns radians by default. Multiply by 180/π to convert to degrees, or use the angle unit selector here to switch automatically.

Real-world applications

  • Carpentry and roofing use leg ratios to set rafter pitch and cut angles for framing.
  • Surveying and navigation use right-triangle angles to convert horizontal and vertical distances into a bearing or elevation angle.
  • Physics and engineering resolve vectors into perpendicular components, then use this same relationship in reverse to find a resultant angle.
  • Trigonometry students use it to check arctan, sine, and cosine calculations by hand against a known triangle.

Frequently Asked Questions

How do you find the angles of a right triangle from its sides?
For a right triangle with legs a and b and the right angle at C, the acute angles satisfy tan(A) = a / b and tan(B) = b / a, so A = arctan(a/b) and B = arctan(b/a). Since the interior angles of any triangle sum to 180 degrees and C is fixed at 90 degrees, A and B always add up to 90 degrees.
How do I find the hypotenuse?
Once you know both legs, the Pythagorean theorem gives the hypotenuse directly: c = the square root of (a squared plus b squared). This calculator computes it automatically alongside the two acute angles and the triangle's area.
What if I only know one leg and the hypotenuse, not both legs?
This calculator expects both legs (a and b). If you only have one leg and the hypotenuse, first find the missing leg using the Pythagorean theorem, b = the square root of (c squared minus a squared), then enter both legs here to get the angles, hypotenuse, and area.
Can I get the angles in radians instead of degrees?
Yes. Switch the angle unit selector to Radians before calculating. The underlying arctan calculation is the same either way; degrees are simply radians multiplied by 180 divided by pi.

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