Triangles are fundamental geometric shapes with numerous important formulas. These formulas help us calculate perimeter, area, and solve for unknown sides and angles. In this presentation, we'll explore the most essential triangle formulas including basic properties, area calculations, and special relationships.
The most basic triangle properties are perimeter and angle sum. Perimeter is simply the sum of all three sides: P equals a plus b plus c. The angle sum property states that all interior angles always add up to 180 degrees. For example, in this right triangle with sides 3, 4, and 5, the perimeter is 12 units.
There are several ways to calculate triangle area. The most common is base times height divided by two. Heron's formula uses all three sides, where s is the semi-perimeter. The SAS formula uses two sides and the included angle. Each method is useful depending on what information you have about the triangle.
The Pythagorean theorem is fundamental for right triangles. It states that a squared plus b squared equals c squared, where c is the hypotenuse. For right triangles, area calculation is simplified to half times leg one times leg two. The classic 3-4-5 triangle demonstrates this perfectly with an area of 6 square units.
For general triangles, we use the Law of Sines and Law of Cosines. The Law of Sines relates sides to opposite angles and is useful for angle-angle-side or angle-side-angle cases. The Law of Cosines generalizes the Pythagorean theorem for any triangle and works for side-angle-side or side-side-side cases. These powerful tools solve any triangle problem.