Triangles: Geometry's Most Fundamental Shape
The triangle is the simplest polygon — three sides and three angles — yet it underpins much of geometry, trigonometry, architecture, engineering, and computer graphics. Every polygon can be divided into triangles, making triangular calculations the foundation of computational geometry. Understanding how to calculate the angles, area, and perimeter of a triangle from its sides is a practical skill with applications ranging from land surveying to structural engineering.
The Law of Cosines (SSS → Angles)
Given three side lengths (SSS — side-side-side), the Law of Cosines calculates each angle. For angle A (opposite side a): cos(A) = (b² + c² − a²) ÷ (2bc). Similarly for B and C. Once two angles are known, the third is 180° − A − B. The law of cosines generalizes the Pythagorean theorem — when angle C is 90°, the formula simplifies to c² = a² + b².
Heron's Formula (Area from 3 Sides)
Heron's formula calculates the area of any triangle given only its three sides, without needing the height. First, compute the semi-perimeter: s = (a + b + c) / 2. Then: Area = √(s × (s−a) × (s−b) × (s−c)). This formula is particularly useful in surveying and navigation where heights are hard to measure but side lengths can be determined by distance measurement.
Triangle Types
Triangles are classified by their angles and their side lengths. By angles: a right triangle has one 90° angle; an acute triangle has all angles less than 90°; an obtuse triangle has one angle greater than 90°. By sides: an equilateral triangle has all three sides equal (and all angles are 60°); an isosceles triangle has exactly two equal sides; a scalene triangle has all three sides different. These properties can be read directly from the calculated angles and compared side lengths.
Triangle Inequality
Not every combination of three positive numbers can form a triangle. The triangle inequality theorem states that the sum of any two sides must be greater than the third side: a + b > c, a + c > b, and b + c > a. If any of these conditions fails, no triangle is possible with those side lengths. For example, sides 1, 2, and 10 cannot form a triangle because 1 + 2 = 3, which is not greater than 10. The calculator validates this before computing.
Results are rounded to two decimal places. For architectural or engineering applications, use precise measurement tools.