Area of Triangles

Geometry · Axiom Academy

Why the area of any triangle is just half the base times the height. 1. A Triangle Is Half a Rectangle Draw any triangle inside the smallest rectangle that contains it. The diagonal cuts that rectangle into two equal triangles — so a single triangle covers exactly half the rectangle. Since a rectangle's area is base times height, a triangle's area is half of that. b = base · h = height (perpendicular to the base) 2. The Height Must Be Perpendicular The height is the perpendicular distance from the base to the opposite vertex — measured at a 90° angle, not along a slanted side. Slide the top vertex sideways and the triangle's shape changes (acute → right → obtuse), but as long as the height stays the same, the area never changes . Height falls inside the triangle The height is not always a side of the triangle. It must be perpendicular (at 90°) to the base. Look for the right-angle mark, or draw a dotted line straight from the top vertex down to the base. 3. Example: Basic Triangle Area Find the area of a triangle with base 10 cm and height 6 cm. 4. Example: Finding the Height A triangle has area 48 m² and base 12 m. Find the height. A right triangle has legs of 5 inches and 12 inches. Find its area. Formula: — half of base times height. Height must be perpendicular to the base (a 90° angle). Any side can be the base — just pair it with the matching height. Right triangles: the two legs are automatically the base and the height.

This is the written version of the interactive lesson above. See the full Geometry course.