Area of Parallelograms and Trapezoids

Geometry · Axiom Academy

LESSON Area of Parallelograms and Trapezoids Extending the rectangle formula to slanted shapes — by cutting, sliding, and averaging. 1. The Parallelogram: a Tilted Rectangle Why does a slanted parallelogram have area A = bh , the same as a rectangle? Cut the triangle off one end and slide it to the other end. The shape becomes a rectangle with the very same base b and height h — so the area never changed. 2. Example: Parallelogram Area Find the area of a parallelogram with base 14 cm and height 9 cm . Apply the formula directly. 3. The Trapezoid: Average of Two Bases A trapezoid has two parallel sides — the bases b_1 and b_2 . Make a second copy , flip it, and join the two together: they form a parallelogram with base (b_1 + b_2) and height h . One trapezoid is exactly half of that. half × (sum of bases) × height Take the average of the two bases, then multiply by the height — same number, different intuition. Find the area of a trapezoid with bases 8 m and 12 m and height 5 m . Substitute into the formula and simplify. 5. Example: Finding a Missing Base A trapezoid has area 84 ft² , height 7 ft , and one base of 10 ft . Find the other base by running the formula backward. All three area formulas are built on base × height . The trapezoid is the general case — when the two bases are equal it collapses back into the parallelogram, and a rectangle is a parallelogram that stands up straight. A = bh — same as a rectangle, since the height must be perpendicular.

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