Solve any triangle by providing 3 known values (at least 1 side). Calculates all sides, angles, area, perimeter, medians, inradius, and circumradius.
The Triangle Calculator is a comprehensive geometric tool used to compute the sides, angles, area, and perimeter of any triangle. Triangles are the most fundamental polygons in geometry, and the mathematical rules governing them are the basis for trigonometry and coordinate geometry. Whether dealing with a right-angled triangle, an equilateral triangle, an isosceles triangle, or a scalene triangle, this calculator simplifies complex trigonometric solving.
For right-angled triangles, calculations are governed by the Pythagorean theorem, which states that the sum of the squares of the two shorter sides equals the square of the hypotenuse (a² + b² = c²). Additionally, basic trigonometric ratios—sine, cosine, and tangent—are used to find relationships between sides and angles. If you know at least one side and one other piece of information (an angle or another side), the calculator can resolve the remaining properties.
For oblique triangles (triangles without a 90-degree angle), advanced formulas like the Law of Sines and the Law of Cosines are applied. The Law of Sines establishes that the ratio of the length of a side to the sine of its opposite angle is constant for all three sides: a/sin(A) = b/sin(B) = c/sin(C). The Law of Cosines generalizes the Pythagorean theorem for any angle: c² = a² + b² - 2ab cos(C). These laws allow the calculator to solve triangles given Side-Angle-Side (SAS), Side-Side-Side (SSS), or Angle-Side-Angle (ASA) inputs.
The area of a triangle can be calculated in several ways depending on the known variables. The classic formula is Area = 1/2 * base * height. However, if the height is unknown but all three sides are given, Heron's Formula is used: Area = √[s(s - a)(s - b)(s - c)], where s is the semi-perimeter: (a + b + c) / 2. This mathematical flexibility is built directly into the triangle calculator.
In civil engineering, land surveying, astronomy, and computer graphics, triangles are used to measure distances that cannot be reached directly (a technique called triangulation). By inputting known benchmarks, users can solve complex structural layouts. The triangle calculator provides instant, precise calculations for these critical applications.
Understanding Triangles
A triangle is a three-sided polygon formed by three vertices connected by line segments called edges. Triangles are classified both by the relative lengths of their sides and by their internal angles. An equilateral triangle has three equal sides and three 60° angles. An isosceles triangle has two equal sides, while a scalene triangle has no equal sides.
Key Properties
- The interior angles always sum to exactly 180° (or π radians).
- An exterior angle equals the sum of the two non-adjacent interior angles.
- The sum of any two side lengths always exceeds the third side (triangle inequality).
- A triangle can have at most one angle that is 90° or greater.
Pythagorean Theorem
For any right triangle, the square of the hypotenuse equals the sum of the squares of the other two sides:
Example: a = 3, c = 5 → b = √(25 − 9) = 4
Law of Sines
The ratio of each side to the sine of its opposite angle is constant across the triangle:
Law of Cosines
Generalizes the Pythagorean theorem to any triangle. Useful for SSS and SAS cases:
Area Formulas
Area = ½ × b × h
Area = ½ × a × b × sin(C)
s = (a + b + c) / 2 → Area = √(s(s−a)(s−b)(s−c))
Medians, Inradius & Circumradius
The median of a triangle connects a vertex to the midpoint of the opposite side. All three medians intersect at the centroid. The inradius is the radius of the largest inscribed circle (incircle), calculated as Area / s. The circumradius is the radius of the circumscribed circle passing through all three vertices, calculated as a / (2·sin(A)).
Inradius = Area / s
Circumradius = a / (2·sin(A))
How it Works & Formula
Computes sides, angles, area, and perimeter of any triangle using the Law of Sines, Law of Cosines, or Heron's Formula.
Practical Examples
Side of 6 units: Area = (√3/4) × 6² ≈ 15.59 square units.
Sides 3 and 4: Hypotenuse = √(3² + 4²) = 5.
Frequently Asked Questions
How many values do I need to solve a triangle?
You need exactly 3 known values, and at least one must be a side length. Valid combinations include SSS, SAS, ASA, AAS, and SSA.
What is the ambiguous case (SSA)?
When two sides and a non-included angle are known, there can be zero, one, or two valid triangles. This calculator returns the primary solution.
Can I use radians?
Yes! Switch the angle unit to radians, and you can enter values like pi/2, pi/3, or any numeric radian value.