A trapezoid’s area is half the sum of its parallel bases multiplied by its height. The crucial measurement is the perpendicular distance between those bases. A sloping side usually cannot replace it, even if that side looks vertical in a sketch.
Why does the formula use the average of the bases?
Call the longer parallel base B, the shorter one b, and their perpendicular separation h. Joining two congruent trapezoids along a nonparallel side forms a parallelogram with base B + b and the same height. Each trapezoid therefore occupies half its area.
The second formula adds all four sides to obtain perimeter. Knowing only the two bases and height does not generally determine both sloping sides. Area and perimeter therefore require different information.
Right, isosceles and scalene trapezoids
A right trapezoid has two right angles, so one nonparallel side also measures the height. An isosceles trapezoid has equal nonparallel sides and equal angles along each base. A scalene trapezoid has unequal nonparallel sides. The area formula remains unchanged for all three.
In British English, the same shape is commonly called a trapezium. Definitions also differ on whether parallelograms are included. Here we discuss the familiar case with one pair of parallel sides and unequal bases; the area calculation also works when the bases are equal.
Worked example with bases 8 cm and 5 cm
Take a height of 3 cm. Add the bases first to get 13 cm, multiply by 3 cm, then halve the resulting 39 square centimetres. Halving only one base would change the expression and give the wrong area.
Recovering a base or the height
Multiply the area by two before isolating the unknown. To find B, divide by h and subtract b. To find h, divide twice the area by the sum of the bases.
With area 30 square centimetres, height 2 cm and shorter base 4 cm, the longer base is 26 cm. With area 100 square centimetres and bases 10 cm and 5 cm, the height is approximately 13.33 cm.
Keep the exact fraction until the final rounding. All given lengths must share one unit. Compare this averaging method with the half-product used for triangle area.
