Sprint 8 Converter + Math
Triangle Area Calculator
Area = 0.5 * base * height.
Area
10.0000
How it works
Triangles are the foundational polygon in geometry, structural engineering, and computer graphics (every 3D mesh is made of triangles). The Triangle Area Calculator supports all major area formulas and computes area from sides (Heron's formula), from base and height, from two sides and an included angle, and from coordinates.
**Methods supported** 1. **Base × Height / 2** — the simplest formula; requires perpendicular height to the base 2. **Heron's Formula** — given all three side lengths: Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 (the semi-perimeter). No need to know any angle. 3. **SAS (Side-Angle-Side)**: Area = (1/2) × a × b × sin(C) — given two sides and the included angle 4. **Coordinate formula**: Area = |x₁(y₂−y₃) + x₂(y₃−y₁) + x₃(y₁−y₂)| / 2 — given three vertex coordinates (x,y)
**Triangle type identification** Given three sides, the calculator also identifies the triangle type: equilateral (all sides equal), isosceles (two sides equal), scalene (all different); and right-angled (if a² + b² = c² holds within floating-point tolerance), acute (all angles < 90°), or obtuse (one angle > 90°).
**Practical applications** Land surveying: irregular plots are decomposed into triangles; areas summed. Structural engineering: triangular trusses are rigid because triangles cannot deform without changing side lengths. Computer graphics: GPU rasterisation converts every polygon to triangles before rendering.
**Triangle inequality** A valid triangle requires each side to be less than the sum of the other two: a < b + c, b < a + c, c < a + b. The calculator validates this and reports if the inputs cannot form a triangle.
Privacy: all computation runs in the browser. No data is transmitted.
Frequently Asked Questions
- Heron's formula calculates triangle area from three side lengths alone, without needing any angle. Step 1: compute the semi-perimeter s = (a + b + c) / 2. Step 2: Area = √(s × (s−a) × (s−b) × (s−c)). Example: triangle with sides 5, 12, 13. s = 15. Area = √(15 × 10 × 3 × 2) = √900 = 30. (This is a 5-12-13 right triangle, so you can verify: ½ × 5 × 12 = 30 ✓.) Heron's formula is useful when you know all three sides from measurements but angles are difficult to determine directly.
- For an equilateral triangle with side length a: Area = (√3/4) × a². This is derived from the standard (base × height / 2) formula: the height of an equilateral triangle = (√3/2) × a (via Pythagorean theorem on the half-triangle). So Area = ½ × a × (√3/2 × a) = (√3/4) × a². Example: equilateral triangle with side 10 cm → Area = (1.732/4) × 100 = 43.3 cm².
- Triangles have unique properties that make them ideal for GPU rendering: (1) Any three non-collinear points define exactly one plane — triangles are always planar, while quadrilaterals with 4 vertices may be non-planar (twisted). (2) Triangles are always convex, simplifying rasterisation algorithms. (3) Triangle meshes can approximate any surface to any desired accuracy by using more/smaller triangles. (4) GPU hardware is optimised specifically for triangle rasterisation — the triangle setup unit is a fundamental GPU component. Every polygon in a 3D scene is decomposed into triangles before rendering.
- Land surveyors use the coordinate (shoelace) formula for polygons: Area = ½ |Σ(xᵢ(yᵢ₊₁ − yᵢ₋₁))|, which decomposes any polygon into triangles from the origin. For irregular plots, surveyors place coordinates at each boundary corner using GPS or total station equipment, then compute the polygon area directly. Alternatively, irregular plots are physically divided into triangles (triangulated irregular network, TIN), each computed via Heron's formula, and summed. GIS software automates this for complex parcel geometries.