Triangle Calculator

What is a Triangle Calculator?

A triangle calculator is a versatile geometry tool that helps you solve any triangle problem by calculating missing sides, angles, area, and perimeter. Whether you know three sides, two sides and an angle, or other combinations, this calculator provides complete triangle solutions.

How to Use This Calculator

1. Select Calculation Mode: Choose from 6 solving methods based on what information you have:

   • SSS (Side-Side-Side): When you know all three sides
   • SAS (Side-Angle-Side): When you know two sides and the included angle
   • ASA (Angle-Side-Angle): When you know two angles and the included side
   • AAS (Angle-Angle-Side): When you know two angles and a non-included side
   • SSA (Side-Side-Angle): When you know two sides and a non-included angle
   • Base × Height: For simple area calculation when you know base and height

2. Enter Known Values: Input the measurements you have available.

3. View Complete Solution: Get all sides, all angles, area, perimeter, and triangle classification.

Triangle Formulas

Area Formulas

Heron's Formula (when you know all three sides):

s = (a + b + c) / 2
Area = √[s(s-a)(s-b)(s-c)]

Where s is the semi-perimeter.

Base × Height Formula:

Area = (base × height) / 2

SAS Formula (two sides and included angle):

Area = (1/2) × a × b × sin(C)

Side and Angle Relationships

Law of Cosines:

c² = a² + b² - 2ab × cos(C)
cos(C) = (a² + b² - c²) / (2ab)

Law of Sines:

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

Angle Sum Property:

A + B + C = 180°

Perimeter

Perimeter = a + b + c

Triangle Types

Classification by Sides

• Equilateral: All three sides are equal
• Isosceles: Two sides are equal
• Scalene: All sides are different

Classification by Angles

• Acute: All angles are less than 90°
• Right: One angle equals 90°
• Obtuse: One angle is greater than 90°

Solving Methods Explained

SSS (Side-Side-Side)

When you know all three sides, use:

1. Heron's formula to find the area
2. Law of Cosines to find each angle
3. Triangle inequality to validate (a+b>c, a+c>b, b+c>a)

SAS (Side-Angle-Side)

When you know two sides and the included angle:

1. Law of Cosines to find the third side
2. Law of Sines to find remaining angles
3. Direct area formula: (1/2)ab×sin(C)

ASA (Angle-Side-Angle)

When you know two angles and the included side:

1. Angle sum property to find third angle
2. Law of Sines to find remaining sides
3. Area using SAS formula

AAS (Angle-Angle-Side)

When you know two angles and a non-included side:

1. Similar to ASA, find third angle first
2. Law of Sines to find remaining sides
3. Calculate area and perimeter

SSA (Side-Side-Angle) - Ambiguous Case

When you know two sides and a non-included angle:

• Warning: This can have 0, 1, or 2 solutions!
• Use Law of Sines carefully
• Check if the given angle is opposite the given side

Real-World Applications

• Construction: Roof framing, truss design, and structural analysis
• Navigation: Triangulation for GPS and maritime navigation
• Surveying: Land measurement and mapping
• Engineering: Stress analysis and force vectors
• Architecture: Building design and space planning
• Computer Graphics: 3D modeling and rendering

Frequently Asked Questions

What's the difference between SSS and SAS?

SSS means you know all three sides, while SAS means you know two sides and the angle between them. SAS requires knowing the specific angle that's between the two known sides.

Why is SSA called the ambiguous case?

SSA can produce two different valid triangles (or sometimes none) because the given side opposite the known angle can "swing" to two different positions while maintaining all the given measurements.

Can any three sides form a triangle?

No! The triangle inequality theorem states that the sum of any two sides must be greater than the third side. For example, sides of 1, 2, and 5 cannot form a triangle because 1+2 is not greater than 5.

What's the easiest way to calculate triangle area?

If you know the base and height, use Area = (base × height) / 2. This is the most straightforward formula. If you only know the sides, use Heron's formula.

How accurate are the angle calculations?

This calculator uses JavaScript's built-in trigonometric functions, providing accuracy to approximately 15-16 significant digits, which is more than sufficient for practical applications.

What units can I use?

The calculator works with any consistent unit of measurement (inches, feet, meters, centimeters, etc.). Just ensure all measurements use the same unit. Angles can be entered in degrees or radians.

Triangle Inequality Theorem

For any valid triangle with sides a, b, and c:

• a + b > c
• a + c > b
• b + c > a

If any of these conditions fail, a triangle cannot be formed.

Common Triangle Properties

Historical Note

Triangle geometry has been studied for over 4,000 years. The ancient Babylonians and Egyptians used triangle calculations for construction and land surveying. Greek mathematicians like Euclid and Pythagoras formalized many triangle theorems we still use today.

Educational Note: This calculator is designed for educational and practical purposes. For critical engineering or construction applications, always verify calculations with professional tools and consult experts.