Perimeter and Area - Area of a Triangle

A triangle is a three-sided polygon, and its area represents the total region enclosed within its three boundaries. To calculate this area, we look at the relationship between its base and its corresponding altitude (height).

Any side of a triangle can be considered as the base. The height, or altitude, is the perpendicular line segment drawn from the opposite vertex to the base. Visually, imagine a triangle sitting on a horizontal line (the base); the height is the straight vertical distance from the highest point (vertex) down to that line, forming a $90^{\circ}$ angle.

The area of a triangle is exactly half the area of a parallelogram that has the same base and height. If you take two identical triangles and flip one to join them along a side, they form a parallelogram, which is why the triangle formula includes a factor of $\frac{1}{2}$.

In a right-angled triangle, the two sides that meet at the $90^{\circ}$ angle (the legs) can be used as the base and the height. Visually, this looks like half of a rectangle divided by a diagonal line.

For an obtuse-angled triangle, the height might fall outside the triangle. To visualize this, you must extend the base line with a dotted line and drop a perpendicular from the top vertex to meet this extended line outside the triangle's body.

The unit for area is always in square units, such as $cm^{2}$, $m^{2}$, or $mm^{2}$. This is because area involves the product of two linear dimensions (base and height).

Triangles that share the same base and are located between the same pair of parallel lines will have the same area because their heights remain constant, even if the triangles have different shapes or 'slants'.