Area of Triangles and Quadrilaterals

Start with the rectangle that boxes in a triangle: draw a triangle, then draw the smallest rectangle that contains it, sharing its base and its height. That rectangle has area base times height. The triangle always takes up exactly half of it, no matter how slanted the triangle looks, because the two leftover corner pieces always fit together to equal the other half. That is where (1/2) x base x height comes from, and it never changes.

A parallelogram is two of those triangles glued together along a diagonal, so its area is just base times height, no fraction needed. A trapezoid is trickier: split it into two triangles sharing the same height, one sitting on each parallel side, and adding their areas gives you height times the average of the two bases, or (1/2)(b1 + b2) x height. A rhombus is a parallelogram turned so its diagonals do the work instead: they cross at right angles and split it into four right triangles, which combine to give (1/2) x diagonal1 x diagonal2. Same base-times-height idea, different packaging.

The one place this trips people up is finding the height itself. Height is always the perpendicular distance from the base to the opposite vertex or side, measured straight down, not along a slanted edge. If the side you are given leans over, you often need to drop an altitude and use the Pythagorean theorem or a given right-triangle piece to find that perpendicular length before you can plug into any formula. This connects directly to surface area and volume problems next, where you will be finding the areas of slanted faces using the same perpendicular-height habit.