Right Triangle Calculator
Calculates the hypotenuse, area, perimeter, and angles of a right triangle from its two legs. Perfect for geometry homework, construction layout, and any design task involving perpendicular measurements.
About this calculator
A right triangle contains one 90° angle. Given the two legs (sides adjacent to the right angle) called a and b, all other properties follow from the Pythagorean theorem and basic trigonometry. Hypotenuse: c = √(a² + b²). Area: A = (a × b) / 2. Perimeter: P = a + b + √(a² + b²). Angle opposite side a: α = arctan(a / b) × (180/π) degrees. The remaining acute angle is simply 90° − α, because all three angles must sum to 180°. These formulas have been known since antiquity and underpin modern surveying, architecture, and engineering. The Pythagorean theorem itself states that the square of the hypotenuse equals the sum of the squares of the other two sides: c² = a² + b².
How to use
Suppose side A = 3 units and side B = 4 units. Hypotenuse: c = √(3² + 4²) = √(9 + 16) = √25 = 5 units. Area: A = (3 × 4) / 2 = 12 / 2 = 6 square units. Perimeter: P = 3 + 4 + 5 = 12 units. Angle α opposite side A: α = arctan(3/4) × (180/π) = arctan(0.75) × 57.296 ≈ 36.87°. The other acute angle is 90° − 36.87° = 53.13°. This is the classic 3-4-5 Pythagorean triple, one of the simplest integer right triangles and widely used in construction to check for square corners.
Frequently asked questions
How do I find the missing side of a right triangle when I only know one leg and an angle?
If you know leg a and angle α (opposite to a), you can find leg b using b = a / tan(α). You can then find the hypotenuse with c = a / sin(α). Alternatively, if you know the hypotenuse c and one angle, the legs are a = c · sin(α) and b = c · cos(α). This calculator currently requires both legs; for angle-based inputs, you would rearrange these SOH-CAH-TOA relationships first.
What is a Pythagorean triple and why is it useful in construction?
A Pythagorean triple is a set of three positive integers (a, b, c) that satisfy a² + b² = c², such as (3, 4, 5), (5, 12, 13), or (8, 15, 17). These triples are useful in construction and carpentry because they allow builders to verify a perfect 90° corner using only a tape measure—no angle-measuring tool required. The 3-4-5 method is especially popular: measure 3 units along one wall, 4 units along the adjacent wall, and if the diagonal is exactly 5 units, the corner is square. Larger multiples like (6, 8, 10) or (9, 12, 15) work equally well.
Why is the area of a right triangle half the product of its two legs?
Every right triangle is exactly half of a rectangle. If you place two identical right triangles together along their hypotenuse, they form a rectangle with sides equal to the two legs a and b, which has area a × b. Since the triangle is half of that rectangle, its area is (a × b) / 2. This relationship holds for all triangles—area = ½ × base × height—but for a right triangle the two legs are conveniently already perpendicular, so they serve directly as base and height without any additional calculation.