ganak.appTriangle Calculator
Free triangle calculator to solve any triangle. Enter 3 known values (sides or angles) to find all missing sides, angles, area, perimeter, height, medians, and more. Supports SSS, SAS, ASA, AAS, and SSA with step-by-step solutions.
Enter triangle values to see results
Provide at least 3 values (with at least 1 side) to solve the triangle
How to Solve a Triangle
Understanding the five cases for solving any triangle
Solving a triangle means finding all unknown sides and angles when you know at least three values (with at least one being a side). The method depends on which combination of values you know.
Side-Side-Side
All three sides known. Use the Law of Cosines to find each angle.
Side-Angle-Side
Two sides and the included angle known. Law of Cosines finds the third side.
Angle-Side-Angle
Two angles and the included side. Find the third angle, then Law of Sines for sides.
Angle-Angle-Side
Two angles and a non-included side. Find the third angle, then Law of Sines.
The Ambiguous Case
Two sides and a non-included angle. May produce zero, one, or two valid triangles depending on the relationship between the known sides and angle.
Note: Three angles alone (AAA) cannot determine a unique triangle — they define the shape but not the size. At least one side is required.
Triangle Formulas
Essential formulas used for solving triangles
Law of Cosines
c² = a² + b² − 2ab·cos(C)
Used for SSS and SAS cases. Generalizes the Pythagorean theorem for non-right triangles.
Law of Sines
a / sin(A) = b / sin(B) = c / sin(C) = 2R
Used for ASA, AAS, and SSA cases. The constant ratio equals the circumscribed circle diameter.
Heron’s Formula
Area = √[s(s−a)(s−b)(s−c)]
Calculates area from three sides alone, where s = (a + b + c) / 2 is the semi-perimeter.
SAS Area Formula
Area = ½ × a × b × sin(C)
Calculates area when you know two sides and the included angle between them.
Altitude (Height)
h = 2 × Area / base
Perpendicular distance from a vertex to the opposite side.
Median
m = ½√(2b²+2c²−a²)
Line from a vertex to the midpoint of the opposite side.
Inradius
r = Area / s
Largest inscribed circle radius
Circumradius
R = a / (2 sin A)
Circumscribed circle radius
Types of Triangles
Classification by sides and by angles
By Sides
Equilateral
All 3 sides equal. All angles are 60°.
Isosceles
Two sides equal. Base angles are equal.
Scalene
All sides different. All angles are different.
By Angles
Acute
All angles less than 90°.
Right
One angle is exactly 90°. Pythagorean theorem applies.
Obtuse
One angle is greater than 90°.
Special Right Triangles
30-60-90 Triangle
Side ratio: 1 : √3 : 2
Example: 5, 8.66, 10
45-45-90 Triangle
Side ratio: 1 : 1 : √2
Example: 5, 5, 7.07
Real-World Applications
Where triangle calculations are used in practice
Construction & Architecture
Roof pitch calculations, triangular supports and trusses, staircase angles, and load distribution all rely on triangle geometry.
Navigation & GPS
GPS uses trilateration to pinpoint locations. Aviation and maritime navigation use triangle calculations for route planning.
Land Surveying
Surveyors use triangulation to measure distances and areas of land parcels and determine property boundaries.
Engineering & Physics
Force vectors, bridge design, structural analysis, and optics all involve decomposing problems into triangle calculations.
Frequently Asked Questions
Common questions about triangle calculations, formulas, and properties