Trapezoid Area — Double It into a Parallelogram
Why is a trapezoid ½(a+b)·h? Flip a copy onto it and you get a parallelogram — so one trapezoid is half.
Problem
Show that a trapezoid with parallel sides a, b and height h has area ½(a+b)·h, by dissection.
Cut · rotate · rearrange
Press Play or move Next one step at a time. Follow the pieces until the answer becomes visible. (Arrow keys and Space also work.)
A trapezoid with parallel sides a, b and height h. Its area?
Why this approach works
Make a pair. Copy the trapezoid, rotate it 180°, and fit it against the slanted side: top and bottom swap to form one straight line, so the two trapezoids become a single parallelogram of base a+b and height h. Its area is (a+b)·h, so one trapezoid is half — ½(a+b)·h.
Proof
A 180°-rotated copy joins to form a parallelogram of base a+b and height h. Two trapezoids make one parallelogram, so each is ½(a+b)·h.
Try it yourself
Shrink the top side a to 0: the trapezoid becomes a triangle and ½(0+b)h = ½bh — the triangle formula.
Related mathematics
Beyond MathVoyage
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