A triangle is a fundamental geometric shape. It is a polygon that has exactly three sides, three vertices, and three angles. The vertices are the corner points where two sides meet, and each side connects two vertices.
Triangles can be classified in different ways. By their sides, we have equilateral triangles with all sides equal, isosceles triangles with two equal sides, and scalene triangles with all different sides. By their angles, we have acute triangles where all angles are less than 90 degrees, right triangles with one 90-degree angle, and obtuse triangles 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. An exterior angle equals the sum of the two non-adjacent interior angles. The area of a triangle can be calculated as one-half times the base times the height.
There are several important formulas for triangles. The most basic area formula is one-half times base times height. We can also calculate area using two sides and the included angle, or using Heron's formula with all three side lengths. The perimeter is simply the sum of all three sides. These formulas are essential tools in geometry and have many practical applications.
Triangles have countless real-world applications. In architecture and construction, triangular trusses provide structural stability for roofs and bridges. In navigation and surveying, triangulation helps determine distances and positions. Computer graphics use triangular meshes for 3D modeling. Engineers rely on triangular frameworks for strength and stability. Triangles are truly fundamental shapes that combine mathematical elegance with practical utility.