Triangle Calculator

How to Use the Triangle Calculator

Getting started is simple:
Enter any three values (with at least one being a side) in the input fields, then click Calculate.
The calculator will instantly work out all unknown sides and angles, identify the triangle type—Equilateral, Isosceles, Scalene, Right, Obtuse, or Acute—and compute properties like area, perimeter, heights, and more.

Understanding Triangles

A triangle is a three-sided polygon—formed by joining three straight lines in a plane, with three angles and three vertices. Triangles are one of the most fundamental shapes in geometry, appearing in everything from construction blueprints to engineering and design.

Triangle Types by Sides

• Equilateral: All sides and angles are equal—each angle is always 60°.
• Isosceles: Two sides (and their opposite angles) are equal; the third is different.
• Scalene: All three sides and all three angles are different.

Triangle Types by Angles

• Right Triangle: One angle is exactly 90°.
• Obtuse Triangle: One angle greater than 90°.
• Acute Triangle: All three angles are less than 90°.

A Historical Perspective

Triangles have a rich history in mathematics, dating back thousands of years.

• Ancient civilizations—Egyptians and Babylonians—used triangles for surveying, astronomy, and construction.
• Greek mathematicians such as Euclid and Pythagoras formalized triangle properties, including the legendary Pythagorean Theorem: a² + b² = c².
• Triangular relationships remain critical in today’s science, engineering, and art.

Triangle Classification & Properties

By Sides:

• Equilateral — all sides (and angles) are the same
• Isosceles — two sides/angles are equal
• Scalene — all sides/angles are different

By Angles:

• Right — one angle is 90°
• Obtuse — one angle is > 90°
• Acute — all angles < 90°

Essential Properties & Formulas:

• Perimeter: Sum of all three sides
• Area: 1/2 × base × height or Heron’s formula:
  Area = √[s(s−a)(s−b)(s−c)]
  where s = [(a + b + c)] / 2
• Law of Sines: a / sin⁡A = b / sin⁡B = c / sin⁡C
• Law of Cosines: c² = a²+b² − 2abcos⁡C
• Pythagorean Theorem: For right triangles, a² + b² = c²

Visual Guide

The tool includes an easy-to-read triangle diagram for visualizing the relationships between sides (aa, bb, cc) and angles (AA, BB, CC).
This helps users easily map their inputs or understand results on the schematic.

What the Triangle Calculator Can Do

• Solve for Unknowns:
  Enter any three values (at least one side). The tool computes every missing side and angle using time-tested geometric and trigonometric formulas.
• Calculate All Triangle Properties:
  • Perimeter
  • Area (all cases, even if you don’t know the height)
  • Heights (altitudes from each vertex)
  • Inradius (radius of inscribed circle)
  • Circumradius (radius of circumscribed circle)
  • Type: Equilateral, isosceles, scalene, right, obtuse, or acute.
• Show Step-by-Step Explanations:
  Explanatory notes appear with each result, so you know which formulas were used.

How the Calculator Works

Flexible Input: 

Any combination of 3 values (at least one side).

Smart Formula Selection:

• Uses the Law of Sines for two angles and a side (ASA or AAS).

• Uses the Law of Cosines for two sides and an included angle (SAS), or all three sides (SSS).

• Ensures the triangle’s angles add up to 180° and sides satisfy the triangle inequality.

Full Error Checking:

Warns if you enter values that don’t create a possible triangle.

Step-by-Step Calculation Example

Example 1: Given two sides and an included angle (SAS)

• Input: a = 8, b = 5, b =  C = 60⁰
• Process:
  1. Calculate the missing side c using the Law of Cosines:
     c = a2 + b2 − 2abcos⁡C
  2. Use Law of Sines to find remaining angles AA and BB
  3. Compute area using Heron’s formula
  4. Find heights, inradius, circumradius, and triangle type

Example 2: Given two angles and one side (ASA/AAS)

• Input: b = 12, B = 40⁰, C = 80⁰
• Process:
  1. Find angle A = 180⁰− B − C = 60⁰
  2. Compute other sides using Law of Sines:
     • a = b[ sin⁡A / sin⁡B],
     • c = b[sin⁡C / sin⁡B]
  3. Automatically get area, perimeter, heights, and more.

Applications & Benefits

• Fast and Accurate: Skip the manual math and get instant, precise results for any triangle.
• All Triangle Types Covered: Handles right, equilateral, isosceles, scalene, acute, and obtuse triangles.
• Educational and Practical:
  • Perfect for math students and teachers for homework, learning, or class demonstrations
  • Indispensable for engineers, architects, builders, designers, and anyone solving real-world triangle problems

User Tips

• Ensure your inputs are valid—at least three values, one of which must be a side.
• Satisfy the triangle inequality (sum of any two sides must be greater than the third), and ensure all angles are positive and sum to 180°.
• The calculator will alert you if the triangle cannot exist.

Why Choose This Triangle Calculator?

• Covers Every Scenario: No matter the triangle, the tool provides all possible properties.
• User-Friendly: Clear input fields, diagram, and easy-to-understand results.
• Reliability for Professional & Academic Use: Trusted formulas, checked for impossible solutions.

Whether for homework, engineering design, construction, or curiosity, this Triangle Calculator is your ultimate companion for working with triangles—fast, error-free, and complete.

Some of the most searched FAQs for the Triangle Calculator Tool