A highway median that is wider at one end than the other is a trapezoid — one pair of sides parallel, the other pair slanting. Its area comes from averaging the two parallel sides (the bases) and multiplying by the height between them: A = ((a + b) / 2) × h. A median 200 feet long, 10 feet wide at one end and 14 at the other, covers ((10 + 14) / 2) × 200 = 2,400 square feet of grass to mow or concrete to pour.

Landscapers pricing oddly shaped lawns, roofers measuring trapezoidal sections, and quilters cutting sashing strips all use it. Type the top base in the Base A (top base) box, the bottom base in the Base B (bottom base) box, and the perpendicular height in the Height (h) box. The Area output does the averaging and multiplying in one step — enter 10, 14, and 200 for the median and read 2,400 square feet directly.

The midsegment shortcut

Every trapezoid has a midsegment, the segment joining the midpoints of its two legs, and its length is exactly the average of the bases: m = (a + b) / 2. The area formula then becomes the simplest possible rectangle formula, area = midsegment × height. For the highway median, the midsegment is (10 + 14) / 2 = 12 feet, and 12 × 200 = 2,400 square feet, the same answer by a shorter road. The midsegment also explains why averaging works at all: cut the trapezoid's overhanging end off and paste it onto the short end, and you get a rectangle whose width is the midsegment and whose height is unchanged. Whenever a problem hands you the midsegment directly, skip the averaging step and multiply straight away.

Worked example: an isosceles trapezoid patio

An isosceles trapezoid has its two legs equal, which makes its height recoverable from the bases and one leg. A patio has bases of 18 feet and 30 feet and legs of 10 feet each. The bases differ by 30 − 18 = 12 feet, and symmetry splits that difference evenly, 6 feet on each side. Each side forms a right triangle with a 6-foot base and a 10-foot hypotenuse, so the height is the square root of (100 − 36), the square root of 64, exactly 8 feet. The area is ((18 + 30) / 2) × 8 = 24 × 8 = 192 square feet of pavers. The perimeter, for the edging order, is 18 + 30 + 10 + 10 = 68 feet. On this calculator, type 18 and 30 as the bases and 8 as the height, and the Area output reads 192 directly; the leg work above is the by-hand step that finds the height when only the legs are known.

Worked example: a trapezoidal garden bed

A garden bed is 12 feet across one end, 20 feet across the other, and 9 feet deep between the parallel sides. The average base is (12 + 20) / 2 = 16 feet, and the area is 16 × 9 = 144 square feet of planting space. Mulch sold in bags covering 10 square feet each means 144 / 10 = 14.4 bags, so 15 bags cover the bed with a little to spare. Notice the trap this example avoids: the slanted sides of this bed are longer than 9 feet, and plugging a slant length in as the height would inflate the area and the mulch order. Height is always the perpendicular distance between the parallel sides, never the slant.

Worked example: a trapezoidal roof plane

A roof plane measures 24 feet along the ridge side, 32 feet along the eave side, and 14 feet of perpendicular slope depth between them. The average base is (24 + 32) / 2 = 28 feet, and the area is 28 × 14 = 392 square feet. Roofing is sold by the square, 100 square feet, so 392 square feet is 3.92 squares; a roofer orders 5 squares to cover cuts, starter courses, and waste. The same arithmetic prices a trapezoidal section of siding, a fabric awning, or any flat shape with one pair of parallel edges: average, multiply, then convert to the selling unit.

Worked example: a trapezoid table top

A craftsman builds a table top 36 inches wide at one end, 48 inches at the other, and 30 inches deep. The average base is (36 + 48) / 2 = 42 inches, and the area is 42 × 30 = 1,260 square inches. Dividing by 144 converts to square feet: 1,260 / 144 = 8.75 square feet of top surface, the number that prices the hardwood and the finish. A right trapezoid, where one leg meets the bases at right angles, works identically: bases of 16 and 26 feet with an 11-foot perpendicular leg give ((16 + 26) / 2) × 11 = 21 × 11 = 231 square feet, and the right-angle leg simply is the height, no triangle solving required.

Running the formula backward

The formula runs in reverse just as cleanly. A trapezoidal lot has an area of 360 square feet and bases of 14 and 22 feet; the average base is (14 + 22) / 2 = 18 feet, so the depth between the bases is 360 / 18 = 20 feet. A flower bed has an area of 315 square feet, one base of 18 feet, and a height of 14 feet; the average base must be 315 / 14 = 22.5 feet, so the missing base is 2 × 22.5 − 18 = 27 feet. The pattern is always the same: undo the multiplication first to recover the average base, then undo the averaging to recover the missing base. These backward problems appear constantly in land records, where the area is surveyed but one boundary length is missing from the deed.

A ladder of trapezoid areas to memorize

A few base-height pairs build fast number sense. Bases 6 and 10 with height 4 give ((6 + 10) / 2) × 4 = 8 × 4 = 32 square units. Bases 8 and 14 with height 5 give 11 × 5 = 55. Bases 12 and 18 with height 7 give 15 × 7 = 105. Bases 20 and 28 with height 10 give 24 × 10 = 240. Bases 30 and 42 with height 15 give 36 × 15 = 540. Sanity-check any new answer against this ladder: an area near these anchors for similar measurements is plausible, while an area off by a factor of two usually means the averaging step was skipped and the bases were simply added.

Angles, perimeter, and the isosceles bonus

In an isosceles trapezoid the base angles come in equal pairs, and angles on the same leg are supplementary, adding to 180 degrees. If one base angle is 72 degrees, its partner on the same base is also 72 degrees, and each angle on the other base is 180 − 72 = 108 degrees. That is all you need to cut the lumber for a trapezoidal frame. The perimeter is just the sum of all four sides, bases plus legs, and it needs the legs measured separately because the area formula never uses them. A general trapezoid with bases 18 and 30 and legs 9 and 13 has a perimeter of 18 + 30 + 9 + 13 = 70 feet, whatever its area turns out to be.

Trapezoid, trapezium, and the British-American swap

Naming causes more confusion here than anywhere else in quadrilateral geometry, and it is worth getting straight. In the United States, a trapezoid is a quadrilateral with at least one pair of parallel sides, which is the shape on this page. In Britain, that same shape is called a trapezium, while a British trapezoid is a quadrilateral with no parallel sides at all, what Americans call a trapezium. The two countries swapped the words. So a British textbook asking for a trapezium area wants exactly this formula, and an American searching for trapezium area lands on the right page for the same reason. When a problem uses either word, check the definition it means by looking for the parallel sides, not by trusting the label alone.

Scaling, units, and five mistakes

Doubling every length quadruples the area, because the formula multiplies two lengths: a bed with bases 12 and 20 and height 9 covers 144 square feet, but doubling to bases 24 and 40 and height 18 covers 576 square feet, four times as much, not twice. Keep units consistent before multiplying: bases in feet with height in inches must be converted first, or the answer is off by a factor of 12. Five mistakes account for most wrong trapezoid areas. First, using a slanted leg as the height. Second, forgetting to average and multiplying the sum of the bases by the height, which doubles the answer. Third, averaging the wrong pair, the legs instead of the parallel sides. Fourth, mixing units between bases and height. Fifth, assuming the shape is a trapezoid when no sides are parallel, in which case the formula does not apply at all and the shape must be split into triangles instead.

Related calculators on this site

If the trapezoid is part of a bigger job, companion tools carry the next step. The triangle area calculator handles the triangular pieces a trapezoid splits into when it is cut by a diagonal. The rhombus calculator covers the slanted four-sided shape whose diagonals, not its bases, drive the area. The right triangle calculator recovers a height from a leg and a base difference, the exact step the isosceles patio example needed. The square calculator handles the special case where all sides are equal and every angle is right. Each one uses the same discipline: name the parallel sides first, measure the perpendicular height, and check the answer against a known anchor.

A classroom note on why averaging works

Students often ask why averaging the bases is legitimate rather than a convenient approximation. The cut-and-paste proof settles it: cut a triangle off the wide end of the trapezoid and move it to the narrow end, and the shape becomes a parallelogram whose base is the average of the two original bases and whose height is unchanged. A parallelogram's area is base times height, so the trapezoid's area is average base times height, exactly, with no approximation anywhere. That is also why the midsegment rectangle is not a trick but the same fact drawn differently: the trapezoid and its midsegment rectangle have identical area by construction.

With the pre-filled defaults — Base A (top base) 5, Base B (bottom base) 7, Height (h) 4 — the Trapezoid Area Calculator returns Area 24.0000 m². Change any input and those results update instantly, so you can see exactly how each number you enter moves the answer.