Triangle Solver & Calculator

Solve triangles using the Law of Cosines and calculate comprehensive triangle properties

Quick answer

Triangle Solver & Calculator

The Law of Cosines relates the sides and angles of any triangle. Given two sides and the included angle, you can find the third side. Heron's formula calculates the area using all three sides.

Formula

c² = a² + b² - 2ab·cos(C) | Area = √(s(s-a)(s-b)(s-c)) | s = (a+b+c)/2

📐 Triangle Input

Enter two sides (a, b) and the angle between them (C) to solve the triangle

💡 Example Problems

Triangle Layout
Side A opposite to angle A
Side B opposite to angle B
Side C opposite to angle C (calculated)
Angle C is between sides A and B

📊 Triangle Solution

📐

Enter two sides and the included angle to solve the triangle

📚 Triangle Properties & Formulas

Law of Cosines
c² = a² + b² - 2ab·cos(C)
Relates sides and angles in any triangle
Heron's Formula
Area = √(s(s-a)(s-b)(s-c)), where s = (a+b+c)/2
Calculates area from three sides
Triangle Types by Angles
• Acute: All angles < 90°
• Right: One angle = 90°
• Obtuse: One angle > 90°
Triangle Types by Sides
• Equilateral: All sides equal
• Isosceles: Two sides equal
• Scalene: All sides different
Circle Properties
• Circumradius: R = abc/(4×Area)
• Inradius: r = Area/semiperimeter
Radii of circumscribed and inscribed circles
Heights (Altitudes)
Height to side = 2×Area/side_length
Perpendicular distance from vertex to opposite side

📐Formula

c² = a² + b² - 2ab·cos(C) | Area = √(s(s-a)(s-b)(s-c)) | s = (a+b+c)/2

💡How it works

The Law of Cosines relates the sides and angles of any triangle. Given two sides and the included angle, you can find the third side. Heron's formula calculates the area using all three sides.

ℹ️ What is Triangle Calculator?

A triangle calculator solves triangles by computing unknown sides, angles, area, and perimeter from any combination of known values. It applies the Pythagorean theorem, the Law of Sines, and the Law of Cosines depending on what information is provided.

📐 Formula

Area = ½ × base × height | c² = a² + b² − 2ab·cos(C)
a, b, c— Side lengths of the triangle
A, B, C— Angles opposite to sides a, b, c
Law of Sines— a/sin(A) = b/sin(B) = c/sin(C)
Heron's Formula— Area = √(s(s−a)(s−b)(s−c)) where s = (a+b+c)/2

✏️ Worked Example

Side a: 5
Side b: 7
Angle C (between a & b): 60°
  1. 1Find side c using Law of Cosines: c² = 5² + 7² − 2(5)(7)cos(60°)
  2. 2c² = 25 + 49 − 70 × 0.5 = 74 − 35 = 39
  3. 3c = √39 ≈ 6.24
  4. 4Area = ½ × a × b × sin(C) = ½ × 5 × 7 × sin(60°) ≈ ½ × 5 × 7 × 0.866 ≈ 15.16
  5. 5Perimeter = 5 + 7 + 6.24 = 18.24
✅ Result: Side c ≈ 6.24 | Area ≈ 15.16 sq units | Perimeter ≈ 18.24

💡 How to Interpret Results

  • ▸The sum of all interior angles of any triangle always equals 180°.
  • ▸The Pythagorean theorem (a²+b²=c²) only applies to right triangles (90° angle).
  • ▸The triangle inequality: any side must be less than the sum of the other two sides.
  • ▸Equilateral triangle: all sides equal, all angles 60°, has maximum area for a given perimeter.
  • ▸Obtuse triangle: one angle > 90°. Acute triangle: all angles < 90°. Right triangle: one angle = 90°.

❓ Frequently Asked Questions

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