Area of Parallelograms and Trapezia
The parallelogram: base × height
A parallelogram looks like a pushed-over rectangle. Its area is the
base times the perpendicular height — the straight-up
distance between the two parallel sides, not the length of the slanted side:
A = b \times h
Why? Cut a right-angled triangle off one slanted end and slide it across to the other end.
Nothing is added or removed, so the area is unchanged — but now the shape is an ordinary
rectangle of width b and height
h, whose area we already know is
b \times h.
The trapezium: average the parallel sides
A trapezium has just one pair of parallel sides, of lengths
a and b, separated by a perpendicular
height h. Its area is the average of the two
parallel sides, multiplied by the height:
A = \tfrac{1}{2}(a+b)\,h
If a = b the two parallel sides are equal and the formula collapses
back to the parallelogram's b \times h.
Both formulas measure the perpendicular height — never the slanted side:
-
a parallelogram has area
A = b \times h, the base times the perpendicular height;
-
a trapezium has area
A = \tfrac{1}{2}(a+b)\,h, the average of its two parallel
sides a and b times the distance
h between them.
See it
Step through the figure: cut the triangle off the parallelogram and slide it across to make a
rectangle, then meet the trapezium with its two parallel sides.