Right Triangle Calculator

Solving Right Triangles

A right triangle has one 90° angle. The longest side (opposite the right angle) is called the hypotenuse.

Pythagorean Theorem

SOH CAH TOA

• SOH: Sin = Opposite / Hypotenuse
• CAH: Cos = Adjacent / Hypotenuse
• TOA: Tan = Opposite / Adjacent

Example: 3-4-5 Triangle

• Sides: 3, 4, 5
• Angles: 36.87°, 53.13°, 90°
• Area: (3 × 4) / 2 = 6

Case 1: Two Legs Known

Use the Pythagorean theorem for the hypotenuse, then inverse tangent for the angles.

Example: legs 3 and 4 → hypotenuse √(9 + 16) = 5, angles arctan(3/4) ≈ 36.87° and arctan(4/3) ≈ 53.13°, area (3 × 4)/2 = 6. The two acute angles always add to 90° — a free check on any solution.

Case 2: Hypotenuse and One Leg

Example (5-12-13 triangle): given hypotenuse c = 13 and leg b = 12:

1. Missing leg: a = √(c² − b²) = √(169 − 144) = √25 = 5.
2. Angle A: arcsin(a/c) = arcsin(5/13) ≈ 22.62°.
3. Angle B: 90° − 22.62° ≈ 67.38°.
4. Area: (5 × 12)/2 = 30.

Case 3: One Side and One Acute Angle

Example: angle A = 30°, hypotenuse c = 10:

1. Leg a: c·sin(A) = 10 × 0.5 = 5.
2. Leg b: c·cos(A) = 10 × (√3/2) ≈ 8.66.
3. Angle B: 90° − 30° = 60°.
4. Area: (5 × 8.66)/2 ≈ 21.65.

Area, Perimeter, and Altitude

For the 3-4-5 triangle: area = 6, perimeter = 12, and the altitude to the hypotenuse is 3 × 4 / 5 = 2.4. The altitude splits the triangle into two smaller right triangles, each similar to the original.

Inradius and Circumradius

The inradius is the radius of the largest circle that fits inside the triangle; the circumradius is the circle through all three vertices (for a right triangle it is simply half the hypotenuse, since the hypotenuse is the circle's diameter). For 3-4-5: r = 12/12 = 1, R = 2.5.

Special Right Triangles

The 30-60-90 triangle is half of an equilateral triangle: the short leg is half the hypotenuse, and the long leg is the short leg times √3.

Pythagorean Triples Worth Knowing

Right Triangles in the Real World

• Squaring a corner: builders check a 90° corner with the 3-4-5 rule — measure 3 units along one edge and 4 along the other; the diagonal must read exactly 5.
• Heights and distances: from 20 m away, a 35° sightline to a treetop means the height above eye level is 20 × tan 35° ≈ 14 m.
• Ramps: rise over run is a tangent ratio; a 1:12 ramp sits at arctan(1/12) ≈ 4.76°.

How to Check Your Answer

• The hypotenuse is the longest side and shorter than the sum of the other two.
• The two acute angles sum to exactly 90°.
• Plugging all three sides back into a² + b² = c² must balance.

Common Mistakes

Which Tool for Which Input

The Geometric Mean Theorems

The altitude to the hypotenuse unlocks two elegant relations. If it splits the hypotenuse into segments p and q:

Check on the 3-4-5 triangle: the altitude h = 2.4 splits the hypotenuse into segments 1.8 and 3.2, and indeed 1.8 × 3.2 = 5.76 = 2.4².

Navigation Example

A ship sails 8 km east and then 6 km north. The straight-line distance home is √(64 + 36) = 10 km, on a bearing of arctan(8/6) ≈ 53.13° west of north. Distance and direction from two legs — a right triangle in disguise.

Three Ways to Find the Area

The 3-4-5 triangle's area is 6 by every route:

• Half the rectangle: (3 × 4)/2 = 6.
• Hypotenuse × altitude: (5 × 2.4)/2 = 6.
• Heron's formula: s = 6, √[6 × 3 × 2 × 1] = √36 = 6.

Angle of Elevation Example

From 50 m away, the angle of elevation to the top of a building is 20°. The height above eye level is 50 × tan(20°) ≈ 18.20 m. Angles of depression work identically — the geometry is the same triangle viewed from the other end.

This Calculator vs the General Triangle Solver

Use this right-triangle calculator whenever one angle is exactly 90°: the Pythagorean theorem and basic trig ratios are faster and less error-prone. Reserve the Law of Sines and Cosines for oblique triangles, where no right angle exists.

Significant Figures and Rounding

Match your answer's precision to the inputs: measurements given to two significant figures (e.g., legs 3.0 and 4.0) justify an answer like 5.0, not 5.00000. Rounding intermediate steps too early is the most common source of drift — keep extra digits until the final answer, then round once.

Calculator Workflow

1. Identify the case: two legs, hypotenuse + leg, or side + angle.
2. Enter values in consistent units — meters with meters, not meters with centimeters.
3. Solve, then verify: angles sum to 90°, hypotenuse is longest, a² + b² = c² balances.
4. Round once at the end, to the precision of your least precise input.