Triangle Calculator (By Sides)
Enter the lengths of the three sides to calculate all properties of the triangle.
Enter Side Lengths
Understanding Triangles: A Foundational Guide
Triangles are one of the most fundamental shapes in geometry. Composed of three sides and three angles, they form the building blocks for more complex polygons and are integral to fields like architecture, engineering, and physics. This Triangle Calculator is a powerful tool designed to take the most basic information—the lengths of the three sides—and derive all other essential properties, from angles to area.
The Mathematics Behind the Tool
Our calculator employs several key geometric theorems to deliver its results. Understanding these principles can deepen your appreciation for the logic at work.
1. The Triangle Inequality Theorem
Before any calculation can be made, we must first determine if the three given side lengths can even form a triangle. The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.
a + c > b
b + c > a
If these conditions are not met, the sides cannot connect to form a closed shape. Our calculator validates this first.
2. Area Calculation using Heron's Formula
When you know all three side lengths, the most elegant way to find the area is with Heron's Formula. This method does not require knowing any angles. It uses the semi-perimeter of the triangle.
- First, calculate the semi-perimeter (s): s = (a + b + c) / 2
- Then, apply Heron's Formula for the Area (K): K = √[s(s-a)(s-b)(s-c)]
3. Angle Calculation using the Law of Cosines
To find the interior angles of the triangle, we use the Law of Cosines. This theorem relates the lengths of the sides to the cosine of one of its angles. We can rearrange the formula to solve for each angle (A, B, and C, opposite sides a, b, and c respectively).
B = arccos((a² + c² - b²) / 2ac)
C = arccos((b² + a² - c²) / 2ba)
Our tool uses these calculations and then converts the results from radians to degrees for ease of understanding. For other advanced trigonometric calculations, you can use our Scientific Calculator.
How We Classify Triangles
Based on the side lengths, a triangle can be classified as:
- Equilateral: All three sides are equal in length (a = b = c). All angles are 60°.
- Isosceles: Exactly two sides are equal in length. The angles opposite these sides are also equal.
- Scalene: All three sides have different lengths. All three angles are also different.
Legal Disclaimer & Terms of Use
For Educational & Estimative Purposes: This tool is designed for educational and general estimation purposes. The calculations are based on standard geometric formulas.
No Liability: We are not liable for any inaccuracies or for decisions made based on the results of this calculator. For engineering, construction, or other critical applications, always consult a qualified professional and verify calculations independently.