Welcome to our lesson on the area of a triangle. The area of a triangle is the amount of space enclosed within its three sides. We can calculate this area using a simple formula: Area equals one half times base times height.
Now let's work through a step-by-step example. First, we identify the base of our triangle, which is 6 units long. Next, we identify the height, which is the perpendicular distance from the top vertex to the base, measuring 4 units. Finally, we apply our formula: Area equals one half times 6 times 4, which gives us 12 square units.
The area formula works for all types of triangles. For right triangles, we can use the two perpendicular sides as base and height directly. The area equals one half times a times b, where a and b are the two legs. For any other triangle, we still use the same formula: one half times base times height, where the height is always perpendicular to the base.
There are other ways to calculate triangle area. Heron's formula uses all three side lengths: first calculate s equals a plus b plus c divided by 2, then the area equals the square root of s times s minus a times s minus b times s minus c. When you know the coordinates of the three vertices, you can use the coordinate formula involving the x and y coordinates of each point.
To summarize what we've learned about triangle area: The fundamental formula is one half times base times height, where the height must always be perpendicular to the base. This formula works for all types of triangles. We also explored alternative methods like Heron's formula and the coordinate method. Understanding triangle area is essential for geometry and has many practical applications in construction, design, and engineering.