A triangle is a fundamental geometric shape. It is a polygon with exactly three sides and three vertices. The three vertices are the corner points where the sides meet. Every triangle has three angles formed where the sides intersect.
Triangles can be classified in different ways. By side lengths, we have equilateral triangles where all three sides are equal, isosceles triangles with two equal sides, and scalene triangles where all sides are different. By angles, triangles can be acute with all angles less than 90 degrees, right triangles with one 90-degree angle, or obtuse with one angle greater than 90 degrees.
Triangles have several important properties. The sum of all interior angles in any triangle is always 180 degrees. The triangle inequality states that the sum of any two sides must be greater than the third side. The longest side is always opposite the largest angle. The area of a triangle equals half the base times the height, and the perimeter is simply the sum of all three sides.
Triangles are central to many important mathematical theorems. The Pythagorean theorem applies to right triangles, stating that the square of the hypotenuse equals the sum of squares of the other two sides. The Law of Sines relates the sides and angles of any triangle. The Law of Cosines generalizes the Pythagorean theorem to all triangles. The triangle inequality theorem states the fundamental constraint on triangle side lengths.
Triangles have countless real-world applications. In architecture and construction, triangular trusses provide structural stability to roofs and bridges because triangles are inherently rigid shapes. In navigation, GPS systems use triangulation with satellites to determine precise locations. Computer graphics rely heavily on triangular meshes for 3D modeling and rendering. Engineers use triangular analysis for force calculations and structural design, making triangles fundamental to modern technology and construction.