Sides, Angles & Area
Last reviewed: April 2026
A right triangle calculator computes the missing sides and angles of a right triangle using the Pythagorean theorem and trigonometric functions. Enter any two known values (sides or angles) to find all remaining measurements, including area and perimeter.
A right triangle has one 90° angle. The side opposite the right angle is the hypotenuse (always the longest side), and the other two sides are called legs. Given any two of the three sides, or one side and one acute angle, you can solve for all remaining parts. The Pythagorean theorem (a² + b² = c²) relates the three sides. Trigonometric ratios relate sides to angles: sin(θ) = opposite/hypotenuse, cos(θ) = adjacent/hypotenuse, tan(θ) = opposite/adjacent. These relationships are the foundation of surveying, navigation, construction, and engineering.
Two right triangles appear so frequently that their ratios are worth memorizing. The 45-45-90 triangle has legs in ratio 1:1:√2 — if each leg is 5, the hypotenuse is 5√2 ≈ 7.07. This triangle appears in every square cut diagonally. The 30-60-90 triangle has sides in ratio 1:√3:2 — if the shortest leg is 4, the longer leg is 4√3 ≈ 6.93 and the hypotenuse is 8. This triangle appears in equilateral triangles cut in half. Builders use 3-4-5 triangles (and multiples like 6-8-10 or 9-12-15) to verify right angles in construction without a protractor. Solve general triangles with our Triangle Calculator.
| Triangle | Angles | Side Ratios | Example |
|---|---|---|---|
| 45-45-90 | 45°, 45°, 90° | 1 : 1 : √2 | 5, 5, 7.07 |
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 | 5, 8.66, 10 |
| 3-4-5 | ~37°, ~53°, 90° | 3 : 4 : 5 | 6, 8, 10 |
A right triangle is a triangle containing exactly one 90-degree angle. The side opposite the right angle is the hypotenuse — always the longest side — while the other two sides are called legs (or catheti). Right triangles are fundamental to geometry, trigonometry, physics, engineering, and architecture because they provide the mathematical foundation for measuring distances, angles, heights, and slopes in both two and three dimensions. Every polygon can be decomposed into right triangles, making them the basic building blocks of computational geometry.
The defining relationship of right triangles is the Pythagorean theorem: a² + b² = c², where a and b are the legs and c is the hypotenuse. This theorem, attributed to the ancient Greek mathematician Pythagoras (though known earlier by Babylonian and Indian mathematicians), is arguably the most important theorem in all of mathematics. It establishes that if you know any two sides of a right triangle, you can calculate the third. It also connects geometry to algebra in a profound way — the theorem essentially states that the area of a square built on the hypotenuse equals the sum of the areas of squares built on the two legs. Our Pythagorean Theorem Calculator focuses specifically on this foundational relationship.
Right triangles give rise to the six trigonometric functions that form the backbone of trigonometry. For a given acute angle θ in a right triangle: sine (sin θ) = opposite/hypotenuse, cosine (cos θ) = adjacent/hypotenuse, and tangent (tan θ) = opposite/adjacent. The reciprocal functions are cosecant (csc θ = 1/sin θ), secant (sec θ = 1/cos θ), and cotangent (cot θ = 1/tan θ). These ratios are constant for a given angle regardless of the triangle's size — all right triangles with a 30-degree angle have the same sine, cosine, and tangent values, whether the triangle has a 1-inch hypotenuse or a 1-mile hypotenuse. This scale-invariance is what makes trigonometry so powerful for indirect measurement.
Two special right triangles have exact trigonometric values that appear frequently in mathematics and should be memorized. The 30-60-90 triangle has sides in the ratio 1 : √3 : 2, so sin 30° = 1/2, cos 30° = √3/2, and tan 30° = 1/√3. The 45-45-90 triangle (isosceles right triangle) has sides in the ratio 1 : 1 : √2, so sin 45° = cos 45° = √2/2 and tan 45° = 1. These values appear constantly in physics, engineering, and advanced mathematics, and the underlying triangles can be derived by bisecting an equilateral triangle (30-60-90) or a square (45-45-90).
Solving a right triangle means finding all unknown sides and angles given sufficient known information. The minimum information needed is one side plus one other measurement (another side or a non-right angle). With two sides known, the third side comes from the Pythagorean theorem, and the angles come from inverse trigonometric functions (arctan, arcsin, or arccos). With one side and one angle known, the other sides come from trigonometric ratios, and the other acute angle is simply 90° minus the known angle (since all angles must sum to 180°).
Area calculations for right triangles are simplified because the two legs are perpendicular, making one the base and the other the height: Area = (1/2) × leg₁ × leg₂. The perimeter is simply the sum of all three sides. The inradius (radius of the inscribed circle) of a right triangle has a simple formula: r = (a + b - c)/2, where a and b are legs and c is the hypotenuse. The circumradius (radius of the circumscribed circle) is even simpler: R = c/2 — the hypotenuse is always a diameter of the circumscribed circle, a beautiful result known as Thales' theorem.
Right triangle calculations are used daily in construction, surveying, navigation, and design. Roof pitch is expressed as the ratio of rise (vertical leg) to run (horizontal leg) — a 6/12 pitch means the roof rises 6 inches for every 12 inches of horizontal distance, creating an angle of arctan(6/12) ≈ 26.6 degrees. Staircase design uses right triangles to calculate rise-to-run ratios within building code requirements (typically 7-8 inch rise and 10-11 inch run, giving comfortable angles of approximately 30-38 degrees). Ladder safety follows the 4:1 rule — for every 4 feet of height, the base should be 1 foot from the wall, creating an approximately 75-degree angle.
In navigation, bearing calculations between two points use right triangles formed by the north-south and east-west distance components. GPS receivers calculate position by solving multiple right triangle problems simultaneously using satellite signal timing. In computer graphics, right triangles underlie rotation matrices, perspective projection, ray tracing, and virtually all 3D rendering calculations. In physics, resolving vectors into perpendicular components (a force into horizontal and vertical parts, a velocity into x and y components) uses right triangle trigonometry as the fundamental technique. Even something as simple as determining how far a shadow extends from a building at a given sun angle is a right triangle problem. For related geometric calculations, explore our Area Converter and Projectile Motion Calculator.
Pythagorean triples are sets of three positive integers (a, b, c) that satisfy a² + b² = c². The most familiar examples are (3, 4, 5), (5, 12, 13), (8, 15, 17), and (7, 24, 25). A primitive Pythagorean triple is one where the three numbers share no common factor greater than 1 — every Pythagorean triple is either primitive or a multiple of a primitive triple (for example, (6, 8, 10) = 2 × (3, 4, 5)). All primitive triples can be generated from the formula a = m² - n², b = 2mn, c = m² + n², where m and n are positive integers with m > n, m and n are coprime, and m and n are not both odd. This formula, known since antiquity, generates every primitive triple exactly once and connects right triangle geometry to deep results in number theory. The (3, 4, 5) triple is practically useful in construction — builders use it to verify right angles by measuring 3 feet along one wall, 4 feet along the other, and confirming the diagonal is exactly 5 feet.
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See also: Pythagorean Theorem Calculator · Triangle Calculator · Angle Converter