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Calculate sides, angles, area, and perimeter of a right triangle using the Pythagorean theorem.
c² = a² + b²
A 3-4-5 right triangle has hypotenuse 5 and area 6.
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Comprehensive triangle calculator solving right triangles and general triangles using SSS, SAS, ASA, AAS methods. Calculate area, perimeter, angles, and advanced properties.
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Geometry & Trigonometry
Solve any right triangle using the Pythagorean theorem, trig ratios, and special angle properties. Calculate sides, angles, area, altitude, circumradius, and inradius instantly.
Pythagorean Theorem
a² + b² = c²
Area
(1/2) × a × b
Sine Rule
sin A = a / c
Circumradius
R = c / 2
✓ Reviewed by the CalculatorApp Mathematics & Geometry Team
A right triangle has exactly one 90° angle. The side opposite the right angle is the hypotenuse (longest side); the other two sides are called legs. Right triangles are the foundation of trigonometry: every angle has unique sine, cosine, and tangent ratios linking it to the side lengths.
Right triangle geometry underlies architecture, surveying, GPS triangulation, structural engineering, computer graphics (rasterization), and satellite navigation. The Pythagorean theorem — perhaps the most famous equation in mathematics — applies exclusively to right triangles.
Hypotenuse
c = √(a² + b²)Given both legs
Missing Leg
a = √(c² − b²)Given hyp + one leg
Altitude to c
h = (a × b) / cCreates two similar Δs
Inradius
r = (a + b − c) / 2Incircle radius
Area
A = (1/2) × a × bProduct of legs ÷ 2
Circumradius
R = c / 2Hypotenuse is diameter
sin A
a / cOpposite / hypotenuse
tan A
a / bOpposite / adjacent
| Name | Angles | Side Ratio | Key Property |
|---|---|---|---|
| 30-60-90 | 30°, 60°, 90° | 1 : √3 : 2 | Equilateral triangle bisected |
| 45-45-90 (Isosceles rt) | 45°, 45°, 90° | 1 : 1 : √2 | Diagonal of a unit square |
| 3-4-5 Pythagorean | ~37°, ~53°, 90° | 3 : 4 : 5 | Most-used in construction |
| 5-12-13 Pythagorean | ~22.6°, ~67.4°, 90° | 5 : 12 : 13 | Next simplest triple |
| 8-15-17 Pythagorean | ~25.4°, ~64.6°, 90° | 8 : 15 : 17 | Popular in tiling problems |
~2000 BC — Babylonian
Plimpton 322 tablet lists Pythagorean triples — predating Pythagoras by over 1,000 years.
~600 BC — Indian Vedic
Baudhayana Sulba Sutra states the Pythagorean theorem geometrically for altar construction.
~570–495 BC — Greek
Pythagoras (reportedly) proves the general theorem: a² + b² = c² for all right triangles.
~300 BC — Euclid
Formalizes triangle geometry in Elements (Books I and VI), including Pythagorean proof.
~150 AD — Ptolemy
Develops chord tables equivalent to modern sine tables for astronomical calculations.
Modern era — Applied
Right triangle geometry underpins GPS, computer graphics, FEA mesh generation, and satellite navigation.
Authoritative mathematical constants and geometry reference data for science and engineering.
External linkFree trig and geometry lessons covering SOHCAHTOA, unit circle, and triangle solving.
External linkDeep mathematical treatment including proofs, formulas, and special triangle properties.
External linkConstruction safety standards applying right-triangle slope calculations to ladder safety.
External linkHow spacecraft trajectory calculations rely on spherical geometry and inverse trig.
External linkFree lecture notes and problem sets on Euclidean geometry and trigonometry.
❌ Myth: The Pythagorean theorem works for all triangles.
✅ Fact: a²+b²=c² applies ONLY to right (90°) triangles. For others, use the law of cosines: c²=a²+b²−2ab·cos(C).
❌ Myth: You always need two sides and an angle to solve a triangle.
✅ Fact: Three sides (SSS) alone are sufficient — use the law of cosines to find all angles.
❌ Myth: 3-4-5 only works in specific units.
✅ Fact: The ratio 3:4:5 works in any consistent unit (cm, m, ft, in) — unit consistency is all that matters.
❌ Myth: Sine and cosine are only for right triangles.
✅ Fact: The law of sines (a/sin A = b/sin B) and cosines extend trigonometry to all triangle types.
Pair the triangle solver with area, volume, and scientific calculators for complete math coverage.