Triangle Area Theorem
📐 Formal Definition
The Triangle Area Theorem states that the area of a triangle is equal to one-half the product of its base and its corresponding height (altitude).
$$ A = \frac{1}{2} \times b \times h $$
where $b$ is the length of any side chosen as the base, and $h$ is the perpendicular distance from that base to the opposite vertex (the altitude).
🔍 Why Does This Formula Work? — The Proof
The key insight is that every triangle is exactly half of a parallelogram. Here's the reasoning:
- Start with any triangle with base $b$ and height $h$.
- Duplicate the triangle and rotate the copy 180°. Attach it to the original along one side.
- The result is a parallelogram with base $b$ and height $h$.
- The area of a parallelogram is $A_{\text{parallelogram}} = b \times h$.
- Since the triangle is exactly half the parallelogram: $A_{\text{triangle}} = \frac{1}{2} \times b \times h$.
This proof works for all triangles — acute, right, and obtuse!