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Triangle Solver

Solve triangles using SSS, SAS, or ASA inputs.

Triangle Inputs

Choose a solving method and enter the known values.

SSS uses a, b, c. SAS uses a, b and angle C. ASA uses angles A, B and side c.

Area: — Perimeter: —
Angle A —
Angle B —
Angle C —

Enter valid side lengths to view the solution.

About this topic

Side-Side-Side (SSS) Solve a triangle given all three side lengths.

When all three sides are known, the triangle is completely determined (if it exists). Angles are found using the Law of Cosines, and area via Heron's formula.

Key formulas & rules

  • •All three sides must be positive
  • •Triangle inequality: a + b > c (and cyclic)
  • •Angles sum to exactly 180°
  • •Law of Cosines: cos(C) = (a² + b² − c²) / 2ab
  • •Heron: Area = √(s(s−a)(s−b)(s−c))
  • •SSS triangles are always unique (if valid)

Examples

Input Output Note
a=3, b=4, c=5 A=36.87°, B=53.13°, C=90° Right triangle
a=5, b=6, c=7 A=44.42°, B=57.12°, C=78.46° Scalene
a=b=c=5 All angles = 60° Equilateral
a=2, b=3, c=10 Invalid Violates inequality

Real-world uses

Surveying Construction Navigation
Laws of Cosines & Sines The tools that connect sides and angles.

Two fundamental trigonometric laws solve any triangle: the Law of Cosines handles side-angle-side and side-side-side problems, while the Law of Sines handles angle-side-angle and angle-angle-side problems.

Key formulas & rules

  • •Cosines: c² = a² + b² − 2ab·cos(C)
  • •Sines: a/sin(A) = b/sin(B) = c/sin(C)
  • •Cosines for SSS or SAS problems
  • •Sines for ASA, AAS, or SSA problems
  • •SSA (ambiguous case) may give 0, 1, or 2 triangles
  • •Law of Cosines reduces to Pythagoras when C = 90°

Examples

Input Output Note
a=8, b=6, C=60° c ≈ 7.21 Law of Cosines
A=30°, a=5, B=45° b ≈ 7.07 Law of Sines
a=3, b=4, C=90° c = 5 Pythagoras
A=60°, B=60°, a=5 b=5 Equilateral

Real-world uses

Triangulation in GPS Force vector analysis Optics
Heron's Formula Area from just the three side lengths.

Heron of Alexandria discovered a way to compute a triangle's area using only the three side lengths — no angles or heights required. It uses the semi-perimeter s = (a + b + c) / 2.

Key formulas & rules

  • •s = (a + b + c) / 2 (semi-perimeter)
  • •Area = √(s(s−a)(s−b)(s−c))
  • •Works for any valid triangle
  • •Equivalent to Law of Cosines + ½ab·sin(C)
  • •Generalizes to cyclic quadrilaterals
  • •Negative under root means invalid triangle

Examples

Input Output Note
a=3, b=4, c=5 Area = 6 s=6
a=5, b=6, c=7 Area ≈ 14.70 s=9
a=1, b=1, c=√2 Area = 0.5 Right isosceles
a=10, b=10, c=10 Area ≈ 43.30 s=15

Real-world uses

Land surveying Architecture CAD software
Triangle Classification Named by angles and sides.

Triangles are classified by both their angle measures and their side lengths. Knowing the classification helps predict properties and shortcuts.

Key formulas & rules

  • •By angles: Acute, Right, Obtuse
  • •By sides: Equilateral, Isosceles, Scalene
  • •Acute: all angles < 90°
  • •Right: one angle = 90° (Pythagoras works)
  • •Obtuse: one angle > 90°
  • •Sum of any two angles < 180°

Examples

Input Output Note
3-4-5 Right scalene Classic
5-5-6 Acute isosceles Mostly acute
5-5-5 Equilateral 60° each
2-3-4 Obtuse scalene One angle > 90°

Real-world uses

Structural analysis Engineering design Education

Modes & formulas

1

SSS (Side-Side-Side)

• Finds the three angles and area of a triangle given three sides. • Enter sides a, b, and c. • The calculator uses the Law of Cosines to find the angles and Heron's formula for the area.
cos(C) = (a² + b² - c²) / 2ab

The inputs must satisfy the Triangle Inequality Theorem. If they don't, the tool will return an error because the sides cannot form a closed triangle.

How to use it

Enter the three side lengths. The calculator will instantly display all three angles, the area, the perimeter, and a visual diagram of the triangle.

Frequently asked questions

Why must the sides obey the triangle inequality?
Because if one side is too long, the other two sides cannot reach each other to form a closed shape.
What is Heron's formula?
Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2 is the semi-perimeter.
What is the Law of Cosines?
A formula used to find an angle or side in any triangle: c² = a² + b² - 2ab·cos(C). It generalizes the Pythagorean theorem.
What is Heron’s formula?
A formula to find the area of a triangle using only its side lengths: Area = √(s(s-a)(s-b)(s-c)), where s is the semi-perimeter.
What is the semi-perimeter?
Half of the triangle’s perimeter, calculated as s = (a + b + c) / 2.
What is an SSS triangle?
A triangle defined by knowing the lengths of all three sides (Side-Side-Side).

Related calculators

Area & Perimeter Calculate area and perimeter for 2D geometric shapes. Pythagorean Theorem Solve for side A, B, or hypotenuse C. Circle Calculator Calculate arc length, sector area, and chord length. Distance & Midpoint Find the distance between two points (x₁,y₁) and (x₂,y₂).
Read the full Triangle Solver guide — examples & 18 FAQs