Triangle Calculator

Understanding Triangles and Geometry Fundamentals

A triangle is a fundamental geometric polygon featuring three distinct vertices connected by three line segments known as edges. Typically, a triangle is identified by its corner vertices—for instance, a triangle with vertices a, b, and c is denoted as $\Delta abc$. Triangles are commonly classified based on their side lengths and interior angles. When all three sides share an equal length, it is known as an equilateral triangle. If only two sides are equal, it forms an isosceles triangle. When all side lengths differ completely, it is classified as a scalene triangle.

Visual markings like tick marks on edges or concentric arcs at angles help indicate equal lengths and measurements. While the default calculator preview might resemble an equilateral shape, entering your custom measurements will instantly update the diagram and results to reflect your specific triangle accurately.

Based on internal angles, triangles fall into two primary groups: right triangles or oblique triangles. A right triangle contains one exact 90° angle, with its longest side (opposite the right angle) called the hypotenuse. Any triangle lacking a 90° angle is an oblique triangle, which can be further categorized as either obtuse (one angle greater than 90°) or acute (all angles under 90°).

Essential Triangle Theorems and Laws

• Angle Limit: A triangle can never have more than one vertex with an internal angle measuring 90° or greater.
• Interior & Exterior Sums: All three interior angles always total 180°. An exterior angle equals the sum of its two non-adjacent interior angles, or simply 180° minus the corresponding vertex angle.
• Triangle Inequality: The combined length of any two sides must always exceed the length of the remaining third side.
• Pythagorean Theorem: Exclusive to right triangles where $a^2 + b^2 = c^2$ (with $c$ as the hypotenuse).

Area Calculation Formulas

Given the length of two sides and the angle between them, the following formulas are used to calculate the area of a triangle (formatted exactly as shown in standard mathematical references):

Alternatively, when all three side lengths are known, Heron's formula can be used:

Medians, Inradius, and Circumradius

Medians: A median connects any vertex to the midpoint of the opposing side. All three medians meet at the triangle's exact centroid.

Inradius & Circumradius: The inradius measures the radius of the largest inscribed circle fitting inside the triangle, whereas the circumradius defines the radius of the circumscribed circle passing through all three vertices.