Find the area and perimeter of any trapezoid from its bases, height, and legs — every step shown. Free, exact, and mobile-friendly.
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A trapezoid (called a trapezium in British English) is a quadrilateral with exactly one pair of parallel sides. Those parallel sides are the bases (a and b), the perpendicular distance between them is the height (h), and the two non-parallel sides are the legs (c and d). Our free Trapezoid Calculator computes the area from the bases and height, and the perimeter when you supply the legs — showing each arithmetic step so you can verify the result by hand.
Trapezoids are everywhere once you notice them: the cross-section of a dam or canal, a flower pot, table tops, handbags, the face of a barn, and highway embankments. In calculus, the trapezoidal rule approximates integrals by slicing curves into trapezoids — one of the most used numerical methods in science and engineering. Any time a shape tapers from one width to another, the trapezoid area formula is the tool you reach for.
This page teaches the trapezoid inside and out. After the calculator you will find usage instructions, a visual explanation of why the area formula works (it is really a rectangle in disguise), the formulas with three worked examples, practical scenarios from construction and gardening, the five errors students make most, the special isosceles case, practice problems with answers, and the eight most-asked trapezoid questions.
The defining feature of a trapezoid is its single pair of parallel sides. (Note: some textbooks use the inclusive definition — “at least one pair” — which makes parallelograms a special type of trapezoid. The exclusive definition used here, “exactly one pair,” is standard in US K–12 education and in most competitions.) The parallel sides are called bases regardless of which is longer; the height is measured perpendicularly between them, and the non-parallel sides are the legs.
The area formula, A = (a + b)/2 × h, has a lovely interpretation: it is the area of a rectangle whose width is the average of the two bases and whose height is h. Imagine slicing the trapezoid horizontally at mid-height — the midsegment (the line joining the midpoints of the legs) has length exactly (a + b)/2, and it runs parallel to the bases. Straightening the trapezoid into a rectangle of that width preserves the area. The calculator’s “midsegment check” verifies this identity on your numbers.
A second way to see the formula: duplicate the trapezoid, rotate the copy 180°, and glue it to the original along one leg. The two trapezoids form a parallelogram with base (a + b) and height h, whose area is (a + b)×h. One trapezoid is half of that — hence the division by 2. This cut-and-paste proof works for any trapezoid, and it is the same trick behind the triangle area formula (a triangle is half a parallelogram too).
The special isosceles trapezoid has its legs equal (c = d) and its base angles equal in pairs; it is symmetric about the perpendicular bisector of the bases, and its diagonals are equal in length. An isosceles trapezoid is cyclic — its four vertices lie on a single circle — a fact used in geometry proofs. The right trapezoid has two adjacent right angles (one leg perpendicular to the bases), making its height equal to that leg: a handy simplification the calculator handles naturally if you enter the perpendicular leg as h.
In calculus, trapezoids do heavy lifting through the trapezoidal rule for numerical integration: the area under a curve between x = a and x = b is approximated by (b − a)/2n times the sum of endpoint-weighted function values — essentially averaging the heights of n trapezoidal slices. Engineers use it for everything from computing distances from velocity data to estimating volumes of earthworks, where cross-sections are literally trapezoidal.
Area = (a + b) ÷ 2 × h Perimeter = a + b + c + d
a, b = the two bases (parallel sides); h = perpendicular height; c, d = legs. The midsegment m = (a+b)/2, so Area = m × h.
A trapezoid has bases a = 10 and b = 6 with height h = 4. Average the bases: (10 + 6) ÷ 2 = 8. Multiply by the height: 8 × 4 = 32 square units. The midsegment check: m = 8, and m × h = 32 ✓. These are the calculator’s defaults — press Calculate to watch the steps. Notice the area (32) sits between the rectangle areas you would get using only the short base (6×4 = 24) or only the long base (10×4 = 40): averaging the bases averages the extremes.
Bases 12 and 8, height 5, legs 6 and 7. Area: (12 + 8) ÷ 2 = 10; 10 × 5 = 50 square units. Perimeter: 12 + 8 + 6 + 7 = 33 units. The perimeter needs all four sides — unlike the area, it cannot be found from bases and height alone, because the leg lengths depend on how the top base is positioned (centered? shifted?). Enter all five numbers in the calculator to see both results with steps.
An isosceles trapezoid has bases 14 and 6, and each leg is 5. First find the height: centering the top base leaves (14 − 6) ÷ 2 = 4 of overhang on each side. Each overhang, the leg, and the height form a right triangle: h² + 4² = 5², so h² = 25 − 16 = 9 and h = 3. Now the area: (14 + 6) ÷ 2 × 3 = 10 × 3 = 30 square units. This two-step pattern — Pythagoras for height, then the area formula — solves most textbook trapezoid problems.
A landscaper designs a trapezoidal flower bed against a fence: the back edge (along the fence) is 9 m, the front edge is 5 m, and the bed extends 3 m out from the fence. How much soil — at 0.25 m depth — is needed? First the area: (9 + 5) ÷ 2 = 7 m average width; 7 × 3 = 21 m². Soil volume: 21 × 0.25 = 5.25 m³. At roughly 1.4 tonnes per cubic metre, that is about 7.4 tonnes of topsoil — the number the delivery quote depends on.
Edging for the bed needs the perimeter. The legs are the slanted sides: with the front centered, each overhang is (9 − 5) ÷ 2 = 2 m, and each leg is √(2² + 3²) = √13 ≈ 3.61 m. Perimeter: 9 + 5 + 3.61 + 3.61 ≈ 21.2 m of edging. Enter a = 9, b = 5, h = 3, c = 3.61, d = 3.61 in the calculator to confirm both figures.
A second scenario: a road embankment cross-section is a trapezoid with a 12 m roadbed on top, 20 m at the base, and 4 m height. Cross-sectional area: (12 + 20) ÷ 2 × 4 = 64 m². For a 500 m stretch, the fill volume is 64 × 500 = 32,000 m³ — the quantity earthworks contractors bid on. Real highway estimates are just this calculation repeated over varying cross-sections (the “average end area” method).
A third from manufacturing: a trapezoidal table top measures 180 cm and 120 cm on its parallel edges with 75 cm between them. Area: (180 + 120) ÷ 2 × 75 = 150 × 75 = 11,250 cm² (1.125 m²). A glass cutter pricing at $90/m² would quote about $101 before edging — and the customer can verify the quote with this page in seconds.
The midsegment (median) of a trapezoid joins the midpoints of the legs. It is parallel to the bases, and its length is exactly the average of the bases: m = (a + b)/2. This is not a coincidence — it is the same averaging the area formula performs, which is why Area = m × h is an exact restatement. The midsegment also bisects the height and both diagonals, making it a useful construction line in proofs: many trapezoid theorems are proved by drawing the midsegment and working with the two smaller trapezoids it creates.
The trapezoidal rule turns this geometry into calculus. To estimate ∫ab f(x)dx, split [a,b] into n strips; each strip under the curve is approximately a trapezoid with parallel sides f(xi) and f(xi+1). Summing gives (h/2)[f(x₀) + 2f(x₁) + … + 2f(xn−1) + f(xn)]. The error shrinks as 1/n², and for smooth functions it is remarkably accurate — GPS receivers and spreadsheet integrations use it constantly.
Trapezoids also tile beautifully: two congruent trapezoids make a parallelogram, and certain isosceles trapezoids tile the plane in brick-like patterns. The cross-sections of pyramids are trapezoids (a frustum’s faces), which is why the ancient Egyptians — computing granary volumes — effectively used the frustum volume formula V = h/3 × (A₁ + A₂ + √(A₁A₂)), a 3D cousin of the trapezoid area formula, as recorded in the Moscow Papyrus around 1850 BCE.
A trapezoid has area 60, height 6, and base a = 14. Find base b. From A = (a+b)/2 × h: 60 = (14+b)/2 × 6, so (14+b)/2 = 10, giving 14 + b = 20 and b = 6. Formulas run backward as easily as forward: isolate the average of the bases first ((a+b)/2 = A/h), then subtract the known base. Try verifying with the calculator by entering a = 14, b = 6, h = 6 — the area should read 60.
A right trapezoid has bases 11 and 7, with the left leg perpendicular to the bases and length 4. Because that leg is perpendicular, it is the height: h = 4, no Pythagoras needed. Area: (11 + 7) ÷ 2 × 4 = 9 × 4 = 36 square units. The remaining leg: overhang = 11 − 7 = 4, so leg = √(4² + 4²) = 4√2 ≈ 5.66. Perimeter: 11 + 7 + 4 + 5.66 ≈ 27.66. Right trapezoids appear in stair and ramp problems, where one side is vertical by construction.
Bases 16 and 10, height 7. Find the area. Answer: 91 square units. ((16+10)/2 = 13; 13×7 = 91.)
An isosceles trapezoid has bases 20 and 12 and height 3. Find each leg. (Hint: overhang = 4.) Answer: 5 each. (√(4²+3²) = 5.)
A trapezoid’s area is 45, its bases are 12 and 6. What is its height? Answer: 5. (Average base 9; 45/9 = 5.)
An irrigation canal is dug with a trapezoidal cross-section: 3 m wide at the bottom, 7 m wide at the water surface, 2 m deep. The wetted cross-sectional area is (3 + 7) ÷ 2 × 2 = 10 m². Over a 2 km canal, the water volume held is 10 × 2000 = 20,000 m³ — the figure the water authority uses to schedule releases.
The trapezoidal shape is no accident: the sloped sides (legs ≈ √(2²+2²) ≈ 2.83 m each) prevent bank collapse, unlike vertical walls. The same geometry sizes the concrete lining: wetted perimeter = 3 + 2.83 + 2.83 = 8.66 m per cross-section, times 2000 m = 17,320 m² of lining. One trapezoid formula priced both the water and the concrete.
A trapezoid has one pair of parallel sides (bases a, b) with perpendicular height h and legs c, d. Area=(a+b)/2×h (equivalently midsegment×height); Perimeter=a+b+c+d. Includes isosceles and right-trapezoid cases and the trapezoidal rule for numerical integration. Source: sigmacalculator.net/math-calculators/trapezoid-calculator/.
Area = (a + b) ÷ 2 × h, where a and b are the two parallel bases and h is the perpendicular height between them. Average the bases, then multiply by the height.
Add all four sides: P = a + b + c + d. Unlike area, the perimeter cannot be found from the bases and height alone — you need both leg lengths.
The segment joining the midpoints of the legs. It is parallel to the bases with length (a+b)/2, so Area = midsegment × height exactly.
Under the inclusive definition (at least one pair of parallel sides), yes. Under the exclusive definition used here (exactly one pair), no. US K–12 curricula typically use the exclusive definition.
For an isosceles trapezoid, center the top base: overhang = (a−b)/2 on each side, then h = √(leg² − overhang²) by the Pythagorean theorem.
In US usage a trapezoid has one pair of parallel sides; in UK usage that shape is called a trapezium, and “trapezoid” means no parallel sides. Check your source’s convention.
A trapezoid with equal legs (c = d). Its base angles are equal in pairs, it is symmetric about the midline, its diagonals are equal, and its vertices lie on a circle.
Dam and canal cross-sections, flower pots, table tops, handbags, embankments, and the trapezoidal rule for numerical integration in calculus and engineering.
Cite this page as: Trapezoid Calculator. Sigma Calculator. Retrieved October 12, 2026, from https://sigmacalculator.net/math-calculators/trapezoid-calculator/
Last updated: October 12, 2026