Triangle Calculators
Explore our complete suite of free triangle calculators. Solve right triangles, Pythagorean hypotenuse, Heron's area formula, and missing angles with live SVG visualizers and step-by-step proofs.
Available Triangle Calculators Solvers
Select a calculator to view formula and step-by-step solution
Right Triangles & Pythagoras
4 calculators
Pythagorean Theorem Calculator
Solve for missing side a, b, or hypotenuse c in right triangles with step-by-step radicals and interactive diagram.
Hypotenuse Calculator (c = √(a² + b²))
Calculate the longest side of a right triangle directly from leg lengths a and b for construction and geometry.
45-45-90 Triangle Calculator
Solve special isosceles right triangles with exact radical ratios (1 : 1 : √2) and side lengths.
30-60-90 Triangle Calculator
Calculate short leg, long leg, and hypotenuse of special right triangles with exact ratios (1 : √3 : 2).
Triangle Area
3 calculators
Triangle Area Calculator (All-in-One)
Calculate triangle area using base and height (1/2·b·h) or two sides and included angle (SAS).
Heron's Formula Calculator (3 Sides Area)
Calculate triangle area when only the 3 side lengths (a, b, c) are known using semi-perimeter s.
Equilateral Triangle Calculator
Calculate area, altitude height, and perimeter of an equilateral triangle from a single side length a.
Oblique & Trigonometric Laws
3 calculators
Right Triangle Solver (Sides & Angles)
Solve all sides, angles (α, β, 90°), area, and perimeter from any 2 known values with interactive SVG drawing.
Law of Sines Calculator (a/sin A = b/sin B)
Solve oblique triangles for missing sides and angles using AAS, ASA, and ambiguous SSA cases.
Law of Cosines Calculator (c² = a² + b² - 2ab cos C)
Solve SSS (3 sides to angles) and SAS triangles step-by-step using the law of cosines.
Properties & Similarity
2 calculators
Triangle Perimeter & Semi-Perimeter
Calculate boundary perimeter (a + b + c), semi-perimeter, and verify the Triangle Inequality Theorem.
Similar Triangles Calculator
Calculate scale factors, missing proportional sides, and area ratios (k²) between two similar triangles.
Mastering Triangles: The Backbone of 2D & 3D Geometry
A triangle is the simplest and most rigid polygon in Euclidean geometry, consisting of three straight line segments connecting three non-collinear vertices. Because a triangle cannot be deformed without breaking or changing the length of its sides, it is structurally the strongest geometric shape known to science, forming the fundamental building block of architecture, bridge trusses, 3D computer graphics meshes, and celestial navigation.
By the Angle Sum Theorem, the sum of all three interior angles in any flat triangle always equals exactly $180^\circ$ ($\alpha + \beta + \gamma = 180^\circ$). Triangles are classified either by their side lengths (Equilateral, Isosceles, Scalene) or by their interior angles (Right, Acute, Obtuse).
Explore Triangle Tools by Geometric Focus
Right Triangles & Pythagorean Theorem
Calculate hypotenuse and leg lengths using $a^2 + b^2 = c^2$, and solve special 45-45-90 and 30-60-90 triangles with exact radical ratios.
Triangle Area Solvers
Calculate area using standard base and height ($1/2 \cdot b \cdot h$), three side lengths via Heron's formula, or equilateral triangle shortcuts.
Oblique Triangles (Sines & Cosines)
Solve non-right triangles using the Law of Sines ($a/\sin A = b/\sin B$) and Law of Cosines ($c^2 = a^2 + b^2 - 2ab\cos C$) for SSS and SAS cases.
Perimeter & Similar Triangles
Calculate boundary perimeters ($a + b + c$), semi-perimeters ($s = P/2$), and scale factors ($k$) between proportional similar triangles.
Master Triangle Formulas Quick Reference
Below is a summary of the fundamental formulas utilized throughout our triangle calculation suite:
| Geometric Theorem | Standard Formula | Primary Application |
|---|---|---|
| Pythagorean Theorem | a² + b² = c² => c = √(a² + b²) | Right Triangles (90°) |
| Base & Height Area | Area = (1/2) × b × h | Known Altitude |
| Heron's Formula Area | A = √[s(s-a)(s-b)(s-c)], s = P/2 | 3 Known Sides (No Height) |
| Equilateral Triangle Area | Area = (√3 / 4) × a² | All 3 Sides Equal |
| Law of Sines | a / sin(A) = b / sin(B) = c / sin(C) | AAS & ASA Triangles |
| Law of Cosines | c² = a² + b² - 2ab cos(C) | SAS & SSS Triangles |
Real-World Applications of Triangle Mathematics
Triangles are not merely abstract textbook problems; their geometric stability makes them critical in everyday engineering:
- Construction & Carpentry (The 3-4-5 Rule): Builders verify that corners form perfect 90° right angles by measuring 3 units along one wall, 4 units along the adjacent wall, and verifying that the diagonal hypotenuse is exactly 5 units long.
- Roof Trusses & Structural Bridges: Bridges and warehouse roofs are built using interconnected triangular trusses (such as Warren and Pratt trusses) because triangles cannot deform without breaking their members.
- GPS Navigation & Triangulation: Smartphones determine your exact geographic location on Earth by measuring distances to three separate GPS satellites and finding their triangle intersection point.
- 3D Video Games & Animation: Modern graphics cards (GPUs) render 3D game worlds by drawing millions of tiny shaded triangles (called polygons) every second.