How this calculator works
A trapezoid has one pair of parallel sides, called the bases. Its area is the average of the two bases times the perpendicular height, which is the same as the midsegment times the height. When you enter all four sides, the calculator drops perpendiculars from the ends of the shorter base and uses the two right triangles at the ends to recover the height. The offset of each leg along the longer base comes from the difference of the squared leg lengths. The isosceles mode makes both legs equal and additionally reports the diagonal, which is found from the height and the midsegment. If the legs cannot reach the opposite base with a positive height, the side combination is rejected.
Formula
Area = ½(a + b)h. Midsegment = ½(a + b). With longer base L, shorter base S, legs c and d: x = ½[(L - S) + (c² - d²)/(L - S)] and h = √(c² - x²). Isosceles diagonal = √(h² + midsegment²).
Worked example
A trapezoid with bases 8 m and 4 m and height 3 m has midsegment 6 m and area 18 m². With legs 5 m and 5 m it is isosceles: x = (8 - 4)/2 = 2 m, height √(25 - 4) = √21 ≈ 4.5826 m, area 6√21 ≈ 27.4955 m² and diagonal √(21 + 36) = √57 ≈ 7.5498 m.
Advertisement
Frequently asked questions
Which side is the height?
The perpendicular distance between the two parallel bases. A slanted leg is not the height, even though it may look similar in a drawing.
What is the midsegment?
The line joining the midpoints of the two legs. It is parallel to the bases and has length equal to the average of the two bases.
Why do the bases need different lengths in some modes?
With equal bases the shape is a parallelogram, and the offset used to find the height divides by zero. Use the parallelogram formula instead when the bases match.
How is the diagonal found in the isosceles mode?
Drop a perpendicular from the top base. The diagonal forms a right triangle whose horizontal side is the midsegment and whose vertical side is the height.