To calculate the area of a triangle, we use the formula: Area equals one-half times the base times the height. The base is any side of the triangle, and the height is the perpendicular distance from the opposite vertex to the base. This perpendicular line forms a right angle with the base.
Let's work through an example. If we have a triangle with a base of 6 units and a height of 4 units, we can calculate its area using our formula. Area equals one-half times the base times the height. So that's one-half times 6 times 4, which equals one-half times 24, giving us 12 square units. The area of the triangle is 12 square units.
When we know all three sides of a triangle but don't have the height, we can use Heron's formula to calculate the area. First, we find the semi-perimeter s, which is half the perimeter, or the sum of all three sides divided by 2. Then, we calculate the area using the formula: Area equals the square root of s times s minus a times s minus b times s minus c, where a, b, and c are the lengths of the three sides. This formula works for any triangle, regardless of its shape.
Let's apply Heron's formula to calculate the area of a triangle with sides of 5, 6, and 7 units. First, we find the semi-perimeter s, which equals the sum of all sides divided by 2. So s equals 5 plus 6 plus 7, divided by 2, which gives us 9. Now we can apply Heron's formula: Area equals the square root of s times s minus a times s minus b times s minus c. Substituting our values, we get the square root of 9 times 4 times 3 times 2, which equals the square root of 216. This gives us approximately 14.7 square units.
To summarize what we've learned about calculating triangle areas: The basic formula is one-half times the base times the height, where the height is the perpendicular distance from a vertex to the opposite side. When we know all three sides but not the height, we can use Heron's formula. First calculate the semi-perimeter s, which is half the perimeter, then apply the formula: Area equals the square root of s times s minus a times s minus b times s minus c. These formulas work for any triangle, regardless of its shape or orientation.