Right Triangle Calculator
Solve a right triangle completely: both legs, the hypotenuse, the two acute angles, and the area. Enter two legs, a leg and the hypotenuse, a leg opposite an acute angle, or the hypotenuse and an acute angle.
a = 3, b = 4, c = 5
- The hypotenuse is c = √(a² + b²) = √(9 + 16) = 5.
- The acute angles: A = arcsin(a/c) ≈ 36.869898°, B = 90° − A ≈ 53.130102°.
- Area = ½ab = 6. Perimeter = 12.
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How to Use the Right Triangle Calculator
Choose the pair of knowns you have. Two legs uses the Pythagorean theorem. A leg and the hypotenuse finds the other leg. A leg opposite A, or the hypotenuse and A, uses sine, cosine, and tangent. Every successful run returns a, b, c, A, B, and the area.
- Pick Two legs, Leg & hypotenuse, Leg a & angle A, or Hypotenuse & angle A.
- Enter two positive values. Angles are in degrees and must be acute (less than 90°).
- Select Solve Triangle and read the sides, angles, area, and the SOH CAH TOA steps.
If you only need one missing side from two known sides, the Pythagorean Theorem Calculator is the shorter page. Area of a non-right triangle belongs on the Triangle Area Calculator. Slope of a line from rise and run belongs on the Slope Calculator.
SOH CAH TOA and the Pythagorean Theorem
In this calculator, angle A is opposite leg a, angle B is opposite leg b, and the right angle sits between the legs. The hypotenuse c is opposite the right angle and is always the longest side.
SOH CAH TOA is the classroom mnemonic: sine is opposite over hypotenuse, cosine is adjacent over hypotenuse, tangent is opposite over adjacent. Once one acute angle is known, the other is 90° minus that angle. The area of a right triangle is always ½ × leg × leg, after the missing leg has been found.
These identities are the right-triangle case of trigonometry in OpenStax Algebra and Trigonometry. They do not apply to an obtuse triangle — there is no 90° angle to sit between two legs.
Right Triangle Calculator Examples
Two legs: 3 and 4
c = √(9 + 16) = 5. A = arcsin(3/5) ≈ 36.8699°. B = 53.1301°. Area = ½ × 3 × 4 = 6.
Leg and hypotenuse: a = 6, c = 10
b = √(100 − 36) = 8. This is the 6-8-10 multiple of 3-4-5. Area = ½ × 6 × 8 = 24.
Leg opposite A: a = 8, A = 30°
sin 30° = 1/2, so c = 8 / (1/2) = 16. tan 30° = 1/√3, so b = 8√3 ≈ 13.856. Area = ½ × 8 × 8√3 = 32√3 ≈ 55.426.
Hypotenuse and A: c = 10, A = 30°
a = 10 sin 30° = 5. b = 10 cos 30° = 5√3 ≈ 8.660. Area = ½ × 5 × 5√3 = 25√3 / 2 ≈ 21.651.
The 5-12-13 triple
c = 13, A = arcsin(5/13) ≈ 22.6199°, B ≈ 67.3801°, area = 30.
Rejected: hypotenuse not longer than the leg
a = 10 and c = 10 cannot be a right triangle with a positive other leg. The calculator reports that the hypotenuse must be longer than the known leg. An angle of 90° or more is rejected the same way: it is not an acute angle in a right triangle.
Which Angle Is A
Angle A is defined to be opposite side a. If a homework figure labels the angle next to leg a, that angle is adjacent to a, not opposite — it is B, not A. Switching them swaps sine and cosine and gives the complementary triangle.
If you know an angle and the adjacent leg instead of the opposite leg, use Two legs after you find the other side on paper, or swap the labels so the known angle is opposite the known leg. This page does not take “adjacent leg and angle” as its own mode; the two angle modes always treat A as opposite a.
Common Right Triangle Mistakes
Using this page on a non-right triangle
SOH CAH TOA and a² + b² = c² require a 90° angle. Other triangles need the law of sines or the law of cosines, which are out of scope here.
Putting the hypotenuse in a leg box
The hypotenuse is the longest side. If you enter it as a leg, the other computed side can be larger than the “hypotenuse,” which is geometrically wrong.
Entering 90° or more for A
The two acute angles of a right triangle are each less than 90°. 90° would duplicate the right angle; more than 90° is obtuse.
Confusing sine and cosine
sin A = opposite / hypotenuse = a/c. cos A = adjacent / hypotenuse = b/c. Mixing them swaps the legs.
Wanting only a missing side
If both knowns are sides and you do not need the angles, the Pythagorean Theorem Calculator is the shorter tool. This page always finishes the triangle.
Right Triangle Calculator FAQs
How do you solve a right triangle?
From two legs, c = √(a² + b²) and A = arcsin(a/c). From a leg and the hypotenuse, the other leg is √(c² − a²). From a side and an acute angle, use sine, cosine, or tangent. The other acute angle is 90° minus A.
What is SOH CAH TOA?
Sine = opposite / hypotenuse, cosine = adjacent / hypotenuse, tangent = opposite / adjacent. On this page, opposite of A is a and adjacent of A is b.
Is this the same as the Pythagorean calculator?
No. Pythagorean finds one missing side from two known sides. This page also finds both acute angles and accepts a side-and-angle pair.
Can the hypotenuse equal a leg?
No. That would force the other leg to be zero, which is not a triangle. The hypotenuse must be strictly longer than each leg.
Does this find the area of any triangle?
No. Area here is ½ab for a right triangle. For Heron or SAS on a general triangle, use the Triangle Area Calculator.
Are my numbers sent to a server?
No. The calculation runs locally in your browser.
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Reviewed by Prof. Dr. Khalil Mudassar (PhD) · Last updated: [mc_modified_date]
Sources
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Disclaimer
This calculator is intended for general educational and practical use. Review important results before relying on them in professional work.