Compute all essential properties of any triangle from three side lengths: area, perimeter, angles, altitudes, medians, circumradius, inradius, and triangle type. Visualize the triangle with labeled sides, angles, and key segments.
The Triangle Solver & Geometry Calculator is a complete interactive tool for analyzing any triangle defined by its three side lengths. Whether you are a student learning Euclidean geometry, a teacher preparing classroom demonstrations, an architect calculating roof trusses, or an engineer performing structural analysis, this tool provides instant, accurate results for all key triangle properties.
From the area (via Heron's formula) and perimeter to the angles (via the Law of Cosines), altitudes, medians, circumradius, and inradius — every metric is computed with high precision. The interactive canvas visualizes the triangle, its sides, angles, altitudes, medians, centroid, and orthocenter, making abstract concepts tangible.
Given sides a, b, c, the semi-perimeter is s = (a + b + c) / 2, and the area by Heron's formula:
Δ = √s(s−a)(s−b)(s−c)
Angles via Law of Cosines: cos A = (b² + c² − a²) / (2bc)
The tool takes three side lengths a, b, and c as input. It first validates the triangle inequality: the sum of any two sides must be greater than the third. If valid, the semi-perimeter s is computed, followed by the area using Heron's formula. Angles are derived from the Law of Cosines, and the remaining properties are calculated using standard geometric formulas:
The triangle type (acute, right, obtuse, equilateral, isosceles, scalene) is determined by comparing the squared side lengths and angle measures. The interactive canvas is drawn by placing vertex A at the origin, vertex B on the positive x‑axis, and computing vertex C's coordinates using the Law of Cosines.
Triangular trusses are fundamental in bridge and roof design. Engineers need to compute the area (for material estimation), angles (for joint fabrication), and altitudes (for load distribution). This calculator provides all these metrics instantly. For example, a symmetrical roof truss with sides 5m, 5m, and 6m has an area of 12 m², an apex angle of approximately 73.7°, and altitudes that determine the placement of vertical supports.
Surveyors use triangulation to measure distances and angles in the field. Given three measured distances between reference points, the triangle solver can compute the interior angles, which are essential for mapping and geolocation. The circumradius and inradius also play roles in optimizing sensor placement and coverage areas.
Triangles are the building blocks of 3D meshes. Developers often need to compute triangle properties for collision detection, rendering optimizations, and texture mapping. This tool provides the area (for face culling), angles (for shading calculations), and medians (for mesh subdivision). The interactive visualization helps in debugging mesh topology.
| Type | Conditions | Properties | Example Sides |
|---|---|---|---|
| Equilateral | a = b = c | All angles 60°, all altitudes equal, centroid = orthocenter = circumcenter = incenter | 3, 3, 3 |
| Isosceles | Two sides equal | Two angles equal, base angles equal, altitude from apex bisects base | 5, 5, 6 |
| Scalene | All sides unequal | All angles unequal, no symmetry | 4, 5, 6 |
| Right | a² + b² = c² (or variant) | One angle = 90°, circumcenter at midpoint of hypotenuse | 3, 4, 5 |
| Acute | a² + b² > c² for all | All angles < 90°, orthocenter inside | 5, 6, 7 |
| Obtuse | a² + b² < c² (for largest side) | One angle > 90°, orthocenter outside | 4, 5, 7 |
The centroid (G) is the intersection of the three medians and is the triangle's center of mass. It divides each median in a 2:1 ratio from the vertex. The orthocenter (H) is the intersection of the altitudes. In acute triangles, both G and H lie inside; in right triangles, H is at the right‑angle vertex; in obtuse triangles, H lies outside. The Euler line connects H, G, and the circumcenter (O), with the property that HG = 2·GO. This tool visualizes both G and H on the canvas, making these abstract concepts concrete.