Right Triangle Calculator

Solve any right triangle by entering any two known values. Calculate all sides, angles, area, and perimeter with step-by-step solutions.

Enter Known Values

Enter any two values to solve the complete triangle:

Side a (opposite angle A):

Side b (opposite angle B):

Hypotenuse c (opposite angle C=90°):

Angle A (degrees):

Angle B (degrees):

Unit:

Right Triangle Diagram

Key Formulas:

Pythagorean: a² + b² = c²
Sine: sin(A) = a/c
Cosine: cos(A) = b/c
Tangent: tan(A) = a/b
Area: A = (a × b) / 2
Perimeter: P = a + b + c

Triangle Solution

3.00 cm

Side a

4.00 cm

Side b

5.00 cm

Hypotenuse c

36.87°

Angle A

53.13°

Angle B

90.00°

Angle C

6.00 cm²

Area

12.00 cm

Perimeter

Solution Steps:

1. Given: Side a = 3 cm, Side b = 4 cm

2. Calculate hypotenuse: c = √(a² + b²) = √(3² + 4²) = √(9 + 16) = √25 = 5.00 cm

3. Calculate angle A: A = arcsin(a/c) = arcsin(3/5) = 36.87°

4. Calculate angle B: B = arcsin(b/c) = arcsin(4/5) = 53.13°

5. Calculate area: Area = (a × b) / 2 = (3 × 4) / 2 = 6.00 cm²

6. Calculate perimeter: P = a + b + c = 3 + 4 + 5 = 12.00 cm

Right Triangle: Quick Solve Framework

1. Core Relationships

a² + b² = c²
sin A = a/c, cos A = b/c, tan A = a/b
A + B = 90° (since C = 90°)
Area = ab/2 ; Perimeter = a + b + c

2. Fast Solve Flow

Two legs? → c via √(a²+b²); then angles via tan⁻¹(a/b).
Leg + hyp? → other leg √(c²−known²).
Leg + acute angle? → use trig ratios; compute c, other leg.
Hyp + angle? → a = c sin A, b = c cos A (or swap).

3. Sanity Checks

c must be longest side.
Angles A,B both < 90° and sum to 90°.
Ratios consistent: (a/c)² + (b/c)² ≈ 1.

4. Special Patterns

3‑4‑5, 5‑12‑13, 8‑15‑17 speed validation; 45‑45‑90 → legs equal; 30‑60‑90 → short:long:hyp = 1:√3:2.

5. Pitfalls

Using degrees vs radians incorrectly in functions.
Mislabeling legs (swap a/b) producing inverted angle.
Hyp not > leg (input error).

6. Micro Examples

a=3 b=4 → c=5, A≈36.87°, B≈53.13°

c=13 a=5 → b=√(169−25)=12

7. Mini FAQ

Two angles only? Need a side to scale triangle.
Precision drift? Round at end; keep internal doubles.
Leg vs opposite? Defined relative to chosen angle.

8. Action Tip

When teaching, derive angles with tan (single ratio) first—reduces early rounding vs chaining sin & arccos.