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Math • Right Triangle Side & Angle Solver

Right Triangle Side & Angle Solver

Solve a right triangle for its hypotenuse, both acute angles, and area from its two legs.

Hypotenuse (c)
5.000
√(a² + b²)
Angle ∠A
36.87°
arctan(a / b)
Angle ∠B
53.13°
90° - ∠A
Triangle Area
6.00
½ × a × b
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Calculation Guide & Reference

Right Triangle Side & Angle Mathematics

Right triangle solver computes the hypotenuse, both acute angles, and area of a right triangle from its two legs.

Standardized Mathematical Formula
c = √(a² + b²) | ∠A = arctan(a / b) | ∠B = 90° − ∠A

The hypotenuse of a right triangle follows the Pythagorean theorem from its two legs. Once the hypotenuse is known, the two acute angles follow from inverse trigonometry — and since a triangle's angles always sum to 180°, the two acute angles of a right triangle must always sum to 90°.

Variables:
a:Leg opposite angle A
b:Leg adjacent to angle A
c:Hypotenuse (opposite the right angle)
How It Works (Step-by-Step)
  • 1Enter the two leg lengths (a and b) of the right triangle.
  • 2View the hypotenuse, both acute angles, and the triangle's area.
Real-World Numerical Example
Legs 3 and 4

A right triangle has legs of length 3 and 4 units.

•Hypotenuse: √(3² + 4²) = √25 = 5 units.
•Angle A: arctan(3/4) ≈ 36.87°.
•Angle B: 90° − 36.87° = 53.13°.
•Area: 0.5 × 3 × 4 = 6 square units.
Result: This is the classic 3-4-5 right triangle, with acute angles of approximately 36.87° and 53.13°, and an area of 6 square units.
Calculation Best Practices & Tips
Examine the discriminant (b² - 4ac) before solving quadratics to instantly know if roots are real, distinct, repeated, or complex.
In statistical analysis, always check both the mean and median to identify if data distribution is skewed by outliers.
When calculating percentage changes, ensure you divide by the original starting baseline value: ((New - Old) / Old) * 100.

Frequently Asked Questions (FAQ)

You can still find the missing leg by rearranging the Pythagorean theorem: the unknown leg equals √(c² − known leg²), then solve for the angles the same way.

The 3-4-5 triangle is just one specific right triangle where all three sides happen to be whole numbers — this solver works for any right triangle, whole-number sides or not.

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