Why You Need a Triangle Calculator
Triangles are fundamental in mathematics, engineering, architecture, and navigation. Solving one — finding all unknown sides, angles, area, and perimeter — can be complex when you lack complete information. Our free triangle calculator handles the math instantly for any type: acute, right, or obtuse.
Types of Triangles and How to Solve Them
By sides: equilateral (all equal), isosceles (two equal), scalene (none equal). By angles: acute (all < 90°), right (one = 90°), obtuse (one > 90°). Depending on known information, there are five solving cases: SSS, SAS, ASA, AAS, and SSA.
The Five Solving Cases
SSS: Know all three sides — use Law of Cosines for each angle. SAS: Two sides and included angle — Law of Cosines for the third side, then Law of Sines for remaining angles. ASA: Two angles and included side — find the third angle (sum = 180°), then Law of Sines. AAS: Similar to ASA. SSA: The trickiest — can produce zero, one, or two solutions (ambiguous case).
Law of Sines and Law of Cosines
Law of Sines: a/sin(A) = b/sin(B) = c/sin(C). Useful for ASA, AAS, and SSA cases. For example, knowing angle A = 40°, side a = 10, and angle B = 60°, you find side b = 10 × sin(60°)/sin(40°) ≈ 13.47. Law of Cosines: c² = a² + b² − 2ab·cos(C). A generalization of the Pythagorean theorem for any triangle, essential for SSS and SAS cases. Our calculator applies both intelligently based on your inputs.
Calculating Area, Perimeter, and Other Properties
Multiple area methods exist: ½ × base × height (when you know base and height), ½ab·sin(C) (two sides and included angle), and Heron's formula (three sides): s = (a+b+c)/2, Area = √(s(s-a)(s-b)(s-c)). Perimeter = a + b + c. Heights (altitudes) from each vertex: h = 2 × Area / opposite side. Medians: m_a = ½√(2b² + 2c² − a²).
Real-World Applications
In construction, triangular roof trusses and bridge supports exploit the triangle's structural strength — unlike other shapes, a triangle cannot deform without changing its side lengths, making it inherently rigid. Surveyors use triangulation to measure distances and map terrain they cannot access directly. Game developers build every 3D model from thousands of triangles, and physics engines use triangle properties for collision detection. Physicists decompose vectors into components using right-triangle trigonometry. Architects calculate roof pitches from span and rise. Our triangle calculator serves students, professionals, and anyone needing quick, accurate solutions.
Frequently Asked Questions
How do I solve a triangle with three sides (SSS)?
Use the Law of Cosines: cos(A) = (b² + c² - a²) / (2bc). Repeat for each angle. Our calculator does this automatically and also computes area using Heron's formula.
What is the Law of Sines?
The ratio of each side to the sine of its opposite angle is constant: a/sin(A) = b/sin(B) = c/sin(C). Useful for ASA, AAS, and SSA cases.
What is the Law of Cosines?
c² = a² + b² - 2ab·cos(C). It generalizes the Pythagorean theorem for any triangle. Essential for SSS and SAS cases.
How do I find the area of a triangle?
Multiple methods: Area = ½ × base × height, or Area = ½ab·sin(C) for two sides and included angle, or Heron's formula for three sides.
Can a triangle calculator handle obtuse triangles?
Yes. The formulas work for acute, right, and obtuse triangles alike. The calculator handles all types and alerts you to invalid inputs.