Triangle Calculator
Free online triangle calculator. Solve SSS, SAS, ASA, AAS, and SSA ambiguous triangles. Calculate sides, angles, area, altitudes, medians, incircle, and circumcircle with interactive SVG visualizers.
Dynamic vector diagram shows sides (a, b, c), angles (α, β, γ), and height in real time.
Enter Dimensions (m)
Support decimalsGeometric Relationship Verification Matrix
Confirmed via Euclidean distance invariants
Centers O, G, H lie on a single collinear ray
Circumradius is at least twice the inradius
Pedagogical Mathematical Breakdowns
Law of Cosines Formulation:
c² = a² + b² - 2ab · cos(γ)
cos(α) = (b² + c² - a²) / (2bc)
Itemized Plaintext Audit Receipt
Cryptographically verified offline calculation audit record
======================================== TRIANGLE CALCULATOR AUDIT RECEIPT Timestamp: 2026-09-04T00:00:00.000Z Solver Mode: SSS ---------------------------------------- DIMENSIONS & ANGLES Side a: 3.0000 m Side b: 4.0000 m Side c: 5.0000 m Angle α (A): 36.8699° Angle β (B): 53.1301° Angle γ (C): 90.0000° ---------------------------------------- AREA & PERIMETER Area (A): 6.0000 sqm Perimeter (P): 12.0000 m Semiperimeter (s): 6.0000 m ---------------------------------------- SECONDARY METRICS Altitudes (ha, hb, hc): 4.0000, 3.0000, 2.4000 m Medians (ma, mb, mc): 4.2720, 3.6056, 2.5000 m Bisectors (ta, tb, tc): 4.2164, 3.3541, 2.4244 m Inradius (r): 1.0000 m Circumradius (R): 2.5000 m Excircles (ra, rb, rc): 2.0000, 3.0000, 6.0000 m ---------------------------------------- CLASSIFICATIONS & THEOREMS Side Classification: Scalene Angle Classification: Right Euler Line Collinearity: VERIFIED (OG:GH = 1:2) Euler Incenter-Circumcenter (dIO): 1.1180 m ========================================
Pythagorean Knotted Ropes (Ancient Egypt)
Ancient Egyptian surveyors ('harpedonaptai') tied 12 equally spaced knots in ropes to construct 3-4-5 right triangles, allowing them to re-establish square property boundaries after annual Nile floods.
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Frequently Asked Questions
The Triangle Inequality Theorem states that for any valid Euclidean triangle, the sum of the lengths of any two sides must be strictly greater than the length of the remaining side (a + b > c, a + c > b, and b + c > a). If any two sides sum to less than or equal to the third side, the segments cannot meet to enclose an area, making a closed triangle physically impossible.
The SSA (Side-Side-Angle) condition occurs when you know two sides and a non-included angle. If the given angle is acute and the opposite side is longer than the perpendicular altitude (h = b·sin(α)) but shorter than the adjacent side (h < a < b), the swinging side can intersect the baseline at two distinct points, creating both a valid acute and a valid obtuse triangle.
Heron's formula computes the area of any triangle directly from its three side lengths a, b, and c using the semi-perimeter s = (a + b + c) / 2. The formula is Area = √(s · (s - a) · (s - b) · (s - c)). This eliminates the need to construct or measure a perpendicular altitude.
In any non-equilateral triangle, the Circumcenter (O), Centroid (G), and Orthocenter (H) are strictly collinear—they lie on a single straight line called the Euler Line. Furthermore, the centroid divides the distance between the circumcenter and orthocenter in an exact 1:2 ratio (OG : GH = 1 : 2). In an equilateral triangle, all four centers collapse into a single point.
An incircle is the largest circle that fits entirely inside the triangle, tangent to all three sides, with inradius r = Area / s and centered at the incenter (intersection of angle bisectors). A circumcircle is the circle that passes through all three triangle vertices, with circumradius R = (abc) / (4·Area) and centered at the circumcenter (intersection of perpendicular bisectors). Euler's inequality proves that R ≥ 2r for all triangles.
No. The Triangle Calculator executes 100% client-side in your browser's memory using native JavaScript and responsive SVG vector rendering. Zero inputs, side lengths, angles, or geometry metrics are ever transmitted to any remote server.