How this calculator works
A trapezoidal prism is a solid whose two ends are identical trapezoids. Its lateral surface wraps around the four side edges, and if the ends are flattened out it forms a rectangle whose width is the trapezoid perimeter and whose height is the prism length. This calculator assumes an isosceles trapezoid, so both sloping sides have the same length. Each is the hypotenuse of a right triangle with the trapezoid height and half the difference between the parallel sides as legs.
Formula
Leg = √(h² + ((a - b)/2)²); perimeter = a + b + 2 × leg; lateral area = perimeter × length; end area = (a + b)/2 × h; total area = lateral + 2 × end area.
Worked example
Parallel sides of 4 m and 2 m with a trapezoid height of 3 m give a sloping side of √(9 + 1) ≈ 3.162 m and a perimeter of about 12.325 m. With a prism length of 10 m, the lateral area is about 123.246 m², each end is 9 m² and the total surface area is about 141.246 m².
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Frequently asked questions
Does this work for a non-isosceles trapezoid?
No. With only the parallel sides and height, the two sloping sides could have different lengths, which changes the perimeter and the lateral area. This calculator uses the isosceles case, where the sloping sides match.
What is the difference between lateral and total area?
Lateral area covers only the four side faces, which is what you would paint on the sides. Total area adds both trapezoidal ends, which is what you need when covering the whole closed solid.
Why is the trapezoid height not the same as the prism length?
The trapezoid height is measured inside one end face, perpendicular to the parallel sides. The prism length runs between the two end faces in the third dimension, at right angles to both.