Area of a Triangle
📐 Formal Definition
The area of a triangle is the measure of the region enclosed by its three sides. It represents the amount of two-dimensional space inside the triangle, measured in square units.
The Standard Formula
$$ A = \frac{1}{2} \times b \times h $$
Where:
- $A$ = Area of the triangle
- $b$ = Base (any side of the triangle)
- $h$ = Height (the perpendicular distance from the base to the opposite vertex)
🔍 Why Does This Formula Work?
Every triangle is exactly half of a parallelogram (or rectangle). Consider this reasoning:
- A rectangle has area $A_{\text{rect}} = b \times h$
- A diagonal of a rectangle divides it into two congruent triangles
- Each triangle has exactly half the area: $A_{\triangle} = \frac{1}{2} \times b \times h$
This relationship holds for all triangles — acute, right, and obtuse — as long as $h$ is the perpendicular height to the chosen base.
📊 Interactive Visualization: Explore the Triangle Area
🖱️ Drag points A, B, and C to see how the area changes! Notice how the height (orange dashed line) is always perpendicular to the base.
🧮 Other Important Area Formulas
Coordinate Geometry Formula
If the vertices are $A(x_1, y_1)$, $B(x_2, y_2)$, $C(x_3, y_3)$:
$$ A = \frac{1}{2} |x_1(y_2 - y_3) + x_2(y_3 - y_1) + x_3(y_1 - y_2)| $$
This is also known as the Shoelace Formula.