How do you solve a triangle with three sides (SSS)?

To solve a triangle with three sides (SSS method), use the Law of Cosines to find the angles: cos(A) = (b² + c² - a²) / (2bc). Then calculate the area using Heron's formula: Area = √[s(s-a)(s-b)(s-c)] where s is the semi-perimeter.

What is the SAS triangle method?

SAS (Side-Angle-Side) method is used when you know two sides and the included angle. Use the Law of Cosines to find the third side: c² = a² + b² - 2ab·cos(C). Then use the Law of Sines to find the remaining angles.

How do you calculate triangle area?

Triangle area can be calculated using several methods: 1) Heron's formula with three sides, 2) ½ × base × height, 3) ½ab·sin(C) with two sides and included angle, or 4) Using coordinates with the shoelace formula.

What makes a valid triangle?

A valid triangle must satisfy the triangle inequality theorem: the sum of any two sides must be greater than the third side. Also, the sum of all angles must equal 180 degrees, and each angle must be positive.

What are the different types of triangles?

Triangles are classified by sides (equilateral, isosceles, scalene) and by angles (acute, right, obtuse). Equilateral triangles have all sides equal, isosceles have two equal sides, and scalene have all different sides.

When do you use the Law of Cosines vs Law of Sines?

Use Law of Cosines when you know: SSS (three sides), SAS (two sides and included angle), or to find angles from sides. Use Law of Sines when you know: ASA, AAS (two angles and a side), or to find sides from angles.

Triangle Calculator - Solve Any Triangle with SSS, SAS, ASA Methods | AI Calculator

Triangles are fundamental geometric shapes with three sides, three angles, and three vertices. Understanding how to solve triangles is essential in mathematics, engineering, architecture, navigation, and many other fields. Our triangle calculator uses proven mathematical methods to find all unknown measurements when given sufficient information.

Triangle Solving Methods

📐 SSS Method

Given: All three sides (a, b, c)

Uses: Law of Cosines + Heron's Formula

Formula: cos(A) = (b² + c² - a²) / (2bc)

Example: Sides 3, 4, 5 → Right triangle

📏 SAS Method

Given: Two sides and included angle

Uses: Law of Cosines + Law of Sines

Formula: c² = a² + b² - 2ab·cos(C)

Example: Sides 3, 4 with 90° angle → Side 5

📐 ASA Method

Given: Two angles and included side

Uses: Law of Sines

Formula: a/sin(A) = b/sin(B) = c/sin(C)

Note: Third angle = 180° - A - B

📏 AAS Method

Given: Two angles and non-included side

Uses: Law of Sines

Similar to ASA: Find third angle first

Then: Use Law of Sines for remaining sides

Triangle Types and Properties

📏 By Sides

Equilateral: All sides equal (a = b = c)
Isosceles: Two sides equal
Scalene: All sides different

📐 By Angles

Acute: All angles < 90°
Right: One angle = 90°
Obtuse: One angle > 90°

🔢 Key Properties

Angle Sum: Always 180°
Triangle Inequality: a + b > c
Largest Angle: Opposite longest side

Area Calculation Methods

Common Area Formulas

1. Heron's Formula (SSS)

Area = √[s(s-a)(s-b)(s-c)]

where s = (a+b+c)/2

2. SAS Formula

Area = ½ab·sin(C)

Two sides and included angle

3. Base × Height

Area = ½ × base × height

When height is known

4. Coordinate Formula

Area = ½|x₁(y₂-y₃) + x₂(y₃-y₁) + x₃(y₁-y₂)|

Using vertex coordinates

Real-World Applications

🏗️ Engineering & Construction

• Structural analysis and truss design
• Roof pitch and rafter calculations
• Bridge and tower construction
• Land surveying and property boundaries

🧭 Navigation & Physics

• GPS triangulation and positioning
• Force vector analysis
• Astronomy and celestial navigation
• Computer graphics and 3D modeling

💡 Tips for Triangle Problem Solving

• Always check if your triangle is valid using the triangle inequality
• Draw a diagram to visualize the problem and label known values
• Choose the most appropriate method based on given information
• Verify your answer by checking that angles sum to 180°
• Use consistent units throughout your calculations
• Round final answers appropriately for your application