Area of Triangles, Parallelograms, and Trapezoids
Cut a triangle from a rectangle and rearrange the pieces. Or shear a rectangle into a parallelogram. Every area formula tells the same story: how to connect back to what we know (area of a rectangle).
Area of a triangle
A triangle is half a parallelogram, which is half... wait, two triangles make a parallelogram. Two parallelograms make a rectangle (if they're the same size). So: Area of triangle = (1/2) × base × height. The height is the perpendicular distance from the base to the opposite vertex, not necessarily a side of the triangle.
Area of a parallelogram
A parallelogram with base b and height h has area b × h. It's the same as a rectangle with the same base and height: if you shear a rectangle (slide the top horizontally), the area stays the same.
Area of a trapezoid
A trapezoid has two parallel sides (bases). Its area is the average of the two bases, times the height: Area = (1/2) × (base₁ + base₂) × height. This comes from the fact that two identical trapezoids make a parallelogram.