Welcome to our lesson on triangle area. The area of a triangle represents the amount of space enclosed by its three sides. Understanding how to calculate triangle area is fundamental in geometry and has many practical applications in engineering, architecture, and everyday problem solving.
The most fundamental formula for calculating triangle area uses the base and height. The area equals one half times base times height. The base can be any side of the triangle, and the height is the perpendicular distance from that base to the opposite vertex. This perpendicular line forms a right angle with the base.
When we know all three sides of a triangle, we can use Heron's formula to find the area. First, we calculate the semi-perimeter s, which is half the sum of all three sides. Then we apply Heron's formula: the area equals the square root of s times s minus a, times s minus b, times s minus c. This formula works for any triangle when all three side lengths are known.
Another useful formula for triangle area uses two sides and the included angle. When we know two sides a and b, and the angle C between them, the area equals one half times a times b times sine of angle C. This formula is particularly helpful in trigonometry problems where angles are given instead of heights.