Right Triangle Calculator

Math

Solve right triangles with instant precision given any two parameters: legs, hypotenuse, angles, or area. Computes sides, angles, area, perimeter, inradius, and all 6 trigonometric ratios with interactive diagrams.

Presets:

💡 Universal Solver: Enter any 2 values below (including at least one length or area). All other sides, angles, altitude, and areas will be calculated automatically.

Length Unit

Side a (vertical leg)

Angle α (opposite to side a)

Side b (horizontal leg)

Angle β (opposite to side b)

Hypotenuse c

Altitude h (perpendicular to hypotenuse)

Area (A)

Right Triangle Solved (γ = 90°)

All Calculated Dimensions (meters (m))

Leg (a) 3 m

Leg (b) 4 m

Hypotenuse (c) 5 m

Altitude (h) 2.4000 m

Area (A) 6 m²

Perimeter (P) 12 m

Inradius (r) 1.0000 m

Circumradius (R) 2.5000 m

Triangle Angles

Angle α (A) 36.87° 0.6435 rad

Angle β (B) 53.13° 0.9273 rad

Angle γ (C) 90.00° π/2 (1.5708 rad)

Trigonometric Functions for Angle α

sin(α) = a/c 0.6000

cos(α) = b/c 0.8000

tan(α) = a/b 0.7500

csc(α) = c/a 1.6667

sec(α) = c/b 1.2500

cot(α) = b/a 1.3333

Hypotenuse Length Across Units

Meters (m) 5

Centimeters (cm) 500

Feet (ft) 16.4042

Inches (in) 196.85

Calculation Steps & Mathematical Working

1 1. Solve Other Legs and Hypotenuse via Angle α

b = a / tan(α) = 3 / tan(36.87°) = 4.0000 m

Tangent is opposite over adjacent: tan(α) = a / b.

2 2. Hypotenuse via Sine

c = a / sin(α) = 3 / sin(36.87°) = 5.0000 m

Sine is opposite over hypotenuse: sin(α) = a / c.

3 Core Geometric Summary

Area = ½ · a · b = 6.0000 m² | Perimeter = a + b + c = 12.0000 m

Inradius r = (a + b - c)/2 = 1.0000 m, Circumradius R = c/2 = 2.5000 m.

4 Altitude to Hypotenuse

h = (a · b) / c = (3.00 × 4.00) / 5.00 = 2.4000 m

Length of perpendicular segment from vertex C (90°) to hypotenuse c.

How This Calculator Works

A right triangle is a triangle in which one interior angle is exactly 90° (a right angle). The side opposite the right angle is the hypotenuse (the longest side), and the other two sides are the legs. Knowing any two independent measurements is sufficient to solve for all remaining sides, angles, altitude, and areas.

📐 Pythagorean Theorem & Right Triangle Trigonometry

a² + b² = c² | sin(α) = a/c | cos(α) = b/c | tan(α) = a/b | Area = ½·a·b = ½·c·h

Hypotenuse (c): The longest side opposite the 90° angle: c = √(a² + b²)

Legs (a, b): The two perpendicular sides adjacent to the right angle: a = c·sin(α), b = c·cos(α)

Altitude to Hypotenuse (h): Length of perpendicular segment from the right angle to the hypotenuse: h = (a × b) / c

Area (A): Half the base times height: Area = (a × b) / 2 = (c × h) / 2

Inradius (r): Radius of the inscribed circle: r = (a + b - c) / 2 = Area / s

Circumradius (R): Radius of the circumscribed circle: R = c / 2

Key Mathematical Rules & Properties

Any 2 Values Principle

A right triangle has 1 fixed angle (90°) and therefore only 2 degrees of freedom. Entering any pair of valid parameters (sides, angles, altitude, or area) uniquely solves the triangle.

Trigonometric Mnemonic SOH-CAH-TOA

Sine = Opposite / Hypotenuse, Cosine = Adjacent / Hypotenuse, Tangent = Opposite / Adjacent.

Altitude Theorem

The altitude to the hypotenuse divides the right triangle into two smaller triangles that are each similar to the original triangle and to each other.

Understanding your calculation

Method, example and limitations

How the calculation works

A right triangle combines a 90° angle with the Pythagorean theorem and trigonometric side ratios to find missing dimensions.

Example to check

Legs 3 and 4 give hypotenuse 5, area 6 and perimeter 12.

Before using the result

Use the correct angle mode and distinguish opposite, adjacent and hypotenuse. Nearly flat triangles can be sensitive to measurement rounding.

Calculation methodology · Report a result that needs review