Welcome to our derivation of the triangle area formula. We'll start by understanding how the area of a rectangle is calculated, which forms the foundation for our triangle formula. A rectangle's area equals base times height.
Now let's see how a right triangle relates to a rectangle. If we take a right triangle and duplicate it, then rotate the copy, we can form a complete rectangle. Since the rectangle is made of two identical triangles, each triangle has half the area of the rectangle.
Now let's extend this concept to any triangle, not just right triangles. We draw a height line from one vertex perpendicular to the opposite side. When we duplicate this triangle and rotate it one hundred eighty degrees, we form a parallelogram with the same base and height.
Now we can complete our derivation. Since the parallelogram is made of two identical triangles, the area of one triangle equals half the parallelogram's area. Therefore, the triangle area formula is one half times base times height. This formula works for any triangle, regardless of its shape.
To summarize what we've learned: We successfully derived the triangle area formula as one half times base times height. This derivation shows how triangles relate to rectangles and parallelograms. The formula works universally for all triangles, making it a fundamental tool in geometry.