What is a Triangle?

A triangle is a polygon with three edges (sides) and three vertices (corners). It is one of the most fundamental shapes in geometry and appears everywhere in nature, architecture, and engineering. The sum of all interior angles in any triangle always equals 180 degrees (π radians).

Triangles are classified by their sides (equilateral, isosceles, scalene) and by their angles (acute, right, obtuse). Understanding these properties is essential for construction, navigation, computer graphics, and countless real-world applications.

Classification by Sides

Equilateral Triangle: All three sides are equal in length, and all three angles are 60°.
Isosceles Triangle: Two sides are equal in length, and the angles opposite those sides are also equal.
Scalene Triangle: All three sides have different lengths, and all three angles are different.

Classification by Angles

Acute Triangle: All three angles are less than 90°.
Right Triangle: One angle is exactly 90°. The side opposite the right angle is called the hypotenuse.
Obtuse Triangle: One angle is greater than 90°.

Triangle Solving Cases

SSS (Side-Side-Side): All three sides are known. Use Law of Cosines to find angles.
SAS (Side-Angle-Side): Two sides and the included angle are known.
ASA (Angle-Side-Angle): Two angles and the included side are known.
AAS (Angle-Angle-Side): Two angles and a non-included side are known.
SSA (Side-Side-Angle): Two sides and a non-included angle. This is the ambiguous case.

Important Theorems

Pythagorean Theorem

For right triangles: a² + b² = c² where c is the hypotenuse.

Law of Sines

a/sin(A) = b/sin(B) = c/sin(C) - relates sides to their opposite angles.

Law of Cosines

c² = a² + b² - 2ab·cos(C) - generalizes the Pythagorean theorem.

Area Calculation Methods

Base × Height Formula

Area = (base × height) / 2 - works when you know a base and its perpendicular height.

Heron's Formula

Area = √[s(s-a)(s-b)(s-c)] where s = (a+b+c)/2 is the semiperimeter.

Trigonometric Formula

Area = (1/2)ab·sin(C) - when you know two sides and the included angle.

Triangle Centers & Radii

Incenter: The center of the inscribed circle (incircle). The inradius r = Area / s.
Circumcenter: The center of the circumscribed circle. The circumradius R = abc / (4·Area).
Centroid: The intersection of the three medians, dividing each median in a 2:1 ratio.
Orthocenter: The intersection of the three altitudes (heights).

Medians and Altitudes

Median

A line segment from a vertex to the midpoint of the opposite side. Formula: ma = (1/2)√(2b² + 2c² - a²)

Altitude (Height)

The perpendicular distance from a vertex to the opposite side. h = 2·Area / base

Real-World Applications

Architecture: Roof trusses, bridges, and structural support systems use triangular designs for stability.
Navigation: Triangulation helps determine position using distances to known reference points.
Computer Graphics: 3D models are composed of triangular polygons (meshes).
Surveying: Land measurement and mapping use triangle calculations extensively.
Physics: Force resolution, vector analysis, and wave mechanics rely on triangle properties.

Triangle Inequality Theorem

For any triangle to exist, the sum of any two sides must be greater than the third side. Mathematically: a + b > c, a + c > b, and b + c > a.

This fundamental rule ensures that the three sides can actually form a closed triangle. If this condition isn't met, the sides cannot connect to form a valid triangle.

Tips for Using This Calculator

Minimum Input: Provide at least 3 values (with at least one side) to solve the triangle.
Angle Units: Choose between degrees or radians using the unit selector.
Auto-Calculate: Results update automatically as you enter values - no button needed.
SSA Ambiguous Case: Be aware that two sides and a non-included angle may have 0, 1, or 2 solutions.
Validation: The calculator checks for valid triangles using the triangle inequality theorem.

Common Mistakes to Avoid

Mixing Units: Don't mix degrees and radians - choose one unit system.
Impossible Triangles: Check that side lengths satisfy the triangle inequality.
Angle Sum: Remember that angles must sum to 180° (or π radians).
SSA Caution: The ambiguous case (SSA) may not have a unique solution.
Precision: For very small or very large triangles, use appropriate precision.

Historical Significance

Triangle geometry has been studied for over 4,000 years. Ancient Egyptians used triangle properties to build the pyramids with remarkable precision. The Pythagorean theorem, discovered around 500 BCE, revolutionized mathematics and remains one of the most important relationships in geometry.

Islamic mathematicians in the medieval period developed sophisticated trigonometry, including the Law of Sines and Law of Cosines, which are essential for solving any triangle. Today, triangle calculations are fundamental to GPS technology, computer graphics, structural engineering, and countless other modern applications.

Triangle Calculator Online Free Tool