Area of a Trapezoid Calculator — Five Input Modes with a Zoomable, Draggable To-Scale Diagram

What is a trapezoid?

A trapezoid is a flat, four-sided shape with exactly one pair of opposite sides parallel. The two parallel sides are called the bases — usually written a and b — and the other two sides are called the legs, written c and d. The perpendicular distance between the bases is the height (or altitude), written h. Its four interior angles always add up to 360°, and the two angles adjacent to the same leg are supplementary.

The naming gets confusing across the Atlantic, and it is worth getting straight before anything else:

So the shape this page calculates — one pair of parallel sides — is a trapezoid in the United States and a trapezium in the UK, Ireland, Australia, New Zealand, India and much of the Commonwealth. The mathematics is identical. If you searched for "area of a trapezium", you are in the right place; the formula is A = ½(a + b)h.

There is a second, subtler disagreement. Some US textbooks use an inclusive definition — a trapezoid has at least one pair of parallel sides — which makes squares, rectangles, rhombuses and parallelograms special cases of trapezoids. Others use an exclusive definition — exactly one pair — which excludes parallelograms. Both definitions give the same area formula, so the calculator works either way. If you enter a shape with equal bases, it will simply tell you it is a parallelogram and still compute the correct area.

The area formula, and why it works

The formula is short enough to memorise in a few seconds:

There are two good ways to see why it is true, and the calculator can show you both.

Why it works, part 1: two trapezoids make a parallelogram

Take a trapezoid and an identical copy rotated 180°. Join the copy along one leg. The two pieces fit together into a single parallelogram. The bottom of that parallelogram is the long base plus the short base — length a + b — and its height is still h. The area of a parallelogram is base × height, so the parallelogram has area (a + b)h. It is made of exactly two trapezoids, so one trapezoid has half that area: ½(a + b)h.

Press ✨ Prove the formula in the tool above and the animation rotates a translucent copy of your own trapezoid 180° around the midpoint of one leg. The copy lands next to the original, the parallelogram outlines itself in cyan, and the labels a + b, h and parallelogram = (a+b)h → ÷ 2 fade in. It is the classic proof performed on your own numbers.

Why it works, part 2: the average-width insight

The quantity (a + b) ÷ 2 is the average of the two bases. It is also the length of the midsegment — the line joining the midpoints of the two legs. That line is parallel to both bases and sits exactly halfway between them. So the formula can be rewritten as:

In other words, a trapezoid has the same area as a rectangle whose width is the average of the two bases and whose height is the same. That is the most useful mental model: a trapezoid is a rectangle with a slanted top, and the slant does not change the area as long as the average width and height stay the same. It is also why sliding the top base sideways — shearing the shape — leaves the area untouched. Turn on the Midsegment toggle and grab the middle of the top edge to shear it.

This is a special case of Cavalieri's principle: two shapes with the same height and the same cross-sectional length at every height have the same area. A sheared trapezoid and a sheared rectangle have identical cross-sections, so their areas match. The same principle, applied vertically, is what lets a stack of trapezoids of different widths approximate the area under any curve — that is the trapezoidal rule of calculus.

Why it works, part 3: the rectangle-plus-two-triangles split

A third way to see it: drop perpendiculars from both top corners to the bottom base. The trapezoid splits into three pieces — a central rectangle of width a and height h, plus two right triangles. The two triangles can always be slid together to form a single rectangle of width (b − a) and height h. Adding the pieces:

Turn on the Split shape toggle in the tool to see the rectangle and the two triangles shaded in different colours.

How to use this calculator

1. Pick the mode that matches what you know. The five tabs across the top are not decoration — they are five genuinely different routes to the same answer. Use Bases & Height if you have the height, Four Sides if you only have the side lengths, Base + Angle if you have a leg and an angle, Coordinates if you have survey points, and Diagonals if you know the two diagonals and the angle between them.
2. Enter your numbers. The diagram redraws to scale immediately. Nothing needs to be submitted and there is no calculate button — every keystroke updates the results, properties, and steps. Use ↑ / ↓ arrow keys in any input field to nudge the value; hold Shift for a bigger step or Alt for a finer one.
3. Choose your units. The length unit applies to every input. The area unit controls the big result card; four other units are shown alongside so you never have to convert by hand.
4. Use reverse mode. In Bases & Height mode and Diagonals mode, the Solve for dropdown lets you enter the area and solve for a missing base, the height, a diagonal, or the angle.
5. Zoom and pan the diagram. Scroll the mouse wheel over the canvas to zoom between 25% and 500%. Drag empty space to pan. Use the + / − / percentage buttons in the bottom-right corner to zoom in, out, or reset to 100%. Double-click the canvas to reset. Mobile users can pinch to zoom and single-finger pan.
6. Drag the shape. In Bases & Height mode the corner handles and the top-edge midpoint handle are all live:
   • B handle (bottom-right) — extends or shortens the bottom base only. The left side stays put.
   • D handle (top-left) — moves the top-left corner. Changes the height and the horizontal offset together.
   • C handle (top-right) — moves the top-right corner. Changes the top base length and the height together.
   • M handle (top edge centre) — shears the shape sideways. The height, both bases and the area all stay the same. This is the Cavalieri demonstration.
   All drags snap to the grid when you release near an intersection. The current value being changed is shown in a small amber tooltip near your cursor.
7. Turn overlays on. Grid, height, midsegment, diagonals, angles and the rectangle-plus-two-triangles split all toggle independently. Labels can be hidden for a clean screenshot.
8. Read the working. The dark panel below the results shows the exact arithmetic with your numbers substituted in, line by line. It is designed for homework and for checking a hand calculation.
9. Estimate a cost. Open the cost estimator to multiply the area by a price per square unit and add a waste allowance — the standard way to budget flooring, paint, roofing or land grading.
10. Export. Copy the results as text, share a link that reproduces your exact inputs, or download the diagram as a PNG or as a scalable SVG.

Five ways to find the area of a trapezoid

Every route below produces the same number. Which one you use depends entirely on which measurements you actually have.

1. Bases and height — the standard formula

A = ½(a + b)h. This is the fastest route and the one taught first. Measure both parallel sides and the perpendicular distance between them. For a = 6 cm, b = 10 cm and h = 4 cm: A = ½ × 16 × 4 = 32 cm².

The only trap is the height. It is not the length of a slanted leg; it is the shortest distance between the two parallel lines. If you measure along a leg, you will overestimate the area by 10–40% depending on how slanted it is.

2. Four sides — no height required

If you know all four sides and you know which pair is parallel, the height is forced. Let b be the longer base, a the shorter base, and c, d the legs. Drop perpendiculars from the top corners to the bottom base. The left leg's horizontal projection is:

For a = 6, b = 10, c = 5, d = 7: x = (16 + 25 − 49) ÷ 8 = −1, h = √(25 − 1) = 4.899, A = ½ × 16 × 4.899 = 39.19 cm². A negative x simply means the top base overhangs to the other side — it is not an error.

Two conditions must hold: the two bases must be different lengths (if a = b the shape is a parallelogram and the height is not fixed by the sides alone), and c² − x² must be positive (otherwise the four lengths cannot close into a trapezoid). The calculator checks both and explains the failure instead of showing a broken number.

3. A base, a leg, and an angle

If you know one base, the adjacent leg, and the interior angle that leg makes with the base, the height follows from a single sine:

For a = 6, b = 10, c = 5 and α = 60°: h = 5 × 0.8660 = 4.330, A = ½ × 16 × 4.330 = 34.64 cm². The cosine of the same angle gives the horizontal projection, which fixes where the top base sits — that is the same x used in the four-sides method.

4. Coordinates — the shoelace formula

If you have the (x, y) coordinates of the four corners, you do not need the height at all. The shoelace (or Gauss) formula gives the area of any simple polygon directly:

For the trapezoid with corners (0, 0), (8, 0), (6, 3), (2, 3), the cross terms are 0, 24, 12 and 0, summing to 36, so A = 18 square units. Check it against the formula: the bases are 8 and 4, the height is 3, and ½ × 12 × 3 = 18. The calculator also verifies that at least one pair of opposite sides is actually parallel, so you are told if the four points form a general quadrilateral rather than a trapezoid.

5. Diagonals and the angle between them

For any convex quadrilateral — trapezoids included — the area is half the product of the diagonals times the sine of the angle between them:

For d₁ = 10, d₂ = 8 and θ = 60°: A = ½ × 10 × 8 × 0.8660 = 34.64 square units. One caution: the diagonals and the angle fix the area but not the shape, so there are infinitely many quadrilaterals with those three measurements. The calculator draws one representative quadrilateral and labels it as illustrative.

Finding the area from four sides — the full explanation

This is the question that sends most people searching, because the standard formula needs a height that is not given. The derivation is worth following once because it shows exactly why the four sides are enough.

Place the longer base on the ground from 0 to b. The shorter base of length a floats at height h, offset horizontally by some unknown amount. The left leg of length c connects (0, 0) to the left end of the top base, and the right leg of length d connects the right end of the top base to (b, 0). Write Pythagoras for both legs and subtract the two equations. The h² terms cancel, leaving a linear equation for the horizontal offset. Solving it gives the formula used by the calculator:

Once h is known, the rest is the standard formula. There is no approximation involved — this is exact, and it is why a trapezoid is the one quadrilateral whose area can be recovered from its four sides alone. For a general quadrilateral, four sides are not enough; you need at least one angle. The trapezoid's parallel constraint supplies that missing information.

The special case of an isosceles trapezoid (c = d) collapses the offset to x = (b − a) ÷ 2, which is the intuitive "the overhang is split evenly between the two sides" result. And because an isosceles trapezoid is cyclic, its area can also be computed with Brahmagupta's formula, A = √((s−a)(s−b)(s−c)(s−d)) with s the semiperimeter — a good cross-check.

One further special case worth knowing: a tangential trapezoid has an inscribed circle touching all four sides, and it exists exactly when a + b = c + d. Its area can also be written A = r × s with r the inradius and s the semiperimeter — one of the rare cases where an area is easier to compute from the perimeter than from the height.

Types of trapezoids

Three named types account for almost every trapezoid you will meet, and each has properties that make some calculations easier.

Isosceles trapezoid

The two legs are equal in length and the two base angles are equal in pairs. It has a vertical line of symmetry through the midpoints of the bases, and its diagonals are equal. Every isosceles trapezoid is cyclic — all four vertices lie on a single circle — which is why Brahmagupta's formula applies. For bases 8 and 12 with legs 5: offset = 2, height = √21 ≈ 4.583, area ≈ 45.83 square units.

Right trapezoid

Two adjacent angles are right angles, so one leg is perpendicular to both bases and its length is the height. That makes it the easiest type to work with: if the vertical leg is c, then A = ½(a + b)c with no trigonometry and no square roots. A right trapezoid is also the shape you get whenever you cut a right triangle off a rectangle.

Scalene trapezoid

No symmetry, no equal legs, no right angles. All four sides and all four angles can differ. It still obeys the same area formula — the formula never cared about symmetry in the first place.

Every trapezoid formula in one place

The trapezoidal rule — where trapezoids meet calculus

The trapezoid formula appears far beyond geometry classrooms: it is the building block of one of the simplest and most widely used numerical integration methods. The trapezoidal rule approximates the area under a curve by slicing it into vertical strips and treating each strip as a trapezoid.

Each strip has parallel vertical sides f(xᵢ) and f(xᵢ₊₁) separated by Δx, so its area is exactly ½(f(xᵢ) + f(xᵢ₊₁))Δx — the trapezoid formula. Summed over all strips and simplified, you get the composite rule above.

The trapezoidal rule is more accurate than the rectangle rule for the same number of strips because it follows the slope of the curve. Its error for a smooth function is roughly proportional to (Δx)² times the second derivative, so halving Δx quarters the error. It is used in engineering for area-under-curve work, in finance for pricing and yield calculations, in physics for work integrals, and in every calculator that numerically integrates a user-entered function.

Higher-order extensions — Simpson's rule, Romberg integration, Gauss–Legendre quadrature — replace trapezoids with curved segments and polynomials, but they are all built on the same idea: approximate a piece of the curve by a simple shape whose area you can compute exactly. The trapezoid is the simplest such shape that follows the slope.

Worked examples

Example 1 — a garden bed

A raised bed is shaped like a trapezoid with parallel sides of 2.4 m and 3.6 m and a depth of 1.5 m. A = ½ × (2.4 + 3.6) × 1.5 = ½ × 6.0 × 1.5 = 4.5 m². If you are buying topsoil to fill it to a depth of 30 cm, the volume is 4.5 × 0.3 = 1.35 m³, which is about 1.35 tonnes of soil at a typical bulk density.

Example 2 — four sides only

A metal plate has parallel sides of 14 cm and 22 cm and legs of 10 cm and 8 cm. x = ((22 − 14)² + 10² − 8²) ÷ (2 × 8) = (64 + 100 − 64) ÷ 16 = 6.25. h = √(100 − 39.06) = 7.808. A = ½ × 36 × 7.808 = 140.5 cm². Notice that no height was measured — it was recovered entirely from the four sides.

Example 3 — a roof gable

A gable end has a 9 ft base, a 3 ft ridge, and a 4 ft rise. A = ½ × (9 + 3) × 4 = 24 ft². If the roof is 30 ft long, the triangular prism of attic space beneath it has a volume of 24 × 30 = 720 ft³.

Example 4 — a land parcel from coordinates

Four survey points are recorded as (0, 0), (34, 0), (28, 17) and (9, 17), in metres. Shoelace: (0×0 − 34×0) + (34×17 − 28×0) + (28×17 − 9×17) + (9×0 − 0×17) = 0 + 578 + 323 + 0 = 901, so A = 450.5 m² — about 0.111 acres. The two horizontal sides are both at constant y, so this really is a trapezoid; the calculator verifies that automatically.

Example 5 — diagonals of a trapezoidal table

A table top has diagonals of 1.8 m and 1.6 m crossing at 72°. A = ½ × 1.8 × 1.6 × sin 72° = ½ × 2.88 × 0.9511 = 1.370 m². This route is handy when the table is already assembled and the diagonals are easier to measure than the height.

Example 6 — embankment volume

A road embankment has a trapezoidal cross-section with a 4 m bottom, a 12 m top, and a 5 m height. The cross-sectional area is ½ × (4 + 12) × 5 = 40 m². If the embankment is 250 m long, the volume of fill is 40 × 250 = 10,000 m³ — roughly 12,500 truckloads of 0.8 m³ each. This is exactly the calculation quantity surveyors use to price earthworks.

Example 7 — a picture mat

A picture frame mat is a trapezoid with a 15 cm top edge, a 25 cm bottom edge, and a 10 cm depth. A = ½ × (15 + 25) × 10 = 200 cm². At a mat price of £0.08 per cm², the mat alone costs £16 — which is why custom-framed pictures with trapezoid mats are more expensive than rectangular ones.

Where trapezoid area shows up in real life

• Roofing and gables. A gable end is a triangle, but the roof plane above a trapezoidal wall section, and the cross-section of a hipped roof, are trapezoids. Roofing quotes are priced per square metre of surface, so the area calculation is the quote.
• Earthworks and road embankments. The cross-section of a road embankment, a drainage ditch, a canal or a railway cutting is a trapezoid. Multiplying that cross-sectional area by the length of the run gives the volume of cut or fill — the standard quantity surveyor's calculation.
• Land and property. Many real plots are trapezoidal because boundaries follow roads that are not parallel. Surveyors compute the area from coordinates with the shoelace formula, exactly as mode 4 does.
• Flooring, tiling and paint. A trapezoidal room or wall needs its area computed before buying materials. Add 10% for cuts and waste, which is what the cost estimator's waste field is for.
• Open-channel hydraulics. Trapezoidal channels are the most common man-made drainage profile because their sloped sides are stable. The flow area is the trapezoid area, and it feeds directly into Manning's equation for flow rate.
• Basketball. Under the international (FIBA) rules used until 2010, the "paint" or key was a trapezoid — roughly 6 m wide at the baseline, 3.6 m at the free-throw line, and 5.8 m deep — giving an area of about 27.8 m². The NBA has always used a rectangle, which is why the two courts look different.
• Architecture and masonry. A keystone at the top of an arch is a trapezoid, wider at the top than the bottom, so that the wedge locks under load. Trapezoidal windows, roof lights and cladding panels are common in modern façades.
• Calculus — the trapezoidal rule. Numerical integration approximates the area under a curve by slicing it into vertical strips and treating each strip as a trapezoid. The composite rule is ∫ ≈ (Δx/2)[f(x₀) + 2f(x₁) + … + 2f(xₙ₋₁) + f(xₙ)], and it is more accurate than a rectangle rule for the same number of strips because it follows the slope of the curve.
• Manufacturing. Die cutters, extrusion profiles, sheet-metal parts and stamped components are often trapezoidal. Cutting die design, material yield calculations, and tolerance analysis all use the trapezoid area formula.
• Cartography. Map projections such as the UTM grid treat small quadrilaterals on the Earth as trapezoids to compute area distortion — a direct application of the four-sides formula when working in projected coordinates.
• Sports fields. Some athletics track inside lanes and field-event sectors (javelin, discus) have trapezoidal markings whose areas are computed for scoring and safety clearances.

Area units and conversions

Area is always a squared unit. If you measured the sides in centimetres, the area is in square centimetres; if you measured in feet, it is in square feet. The conversion factor between two length units is squared when you convert an area — that is why there are 10,763.9 square feet in a square metre, not 3.28.

⚠ Squared conversions surprise people. A square yard is 9 square feet, not 3. A square metre is about 10.76 square feet, not 3.28. Whenever you convert an area, square the length conversion factor. This is the single most common unit error in flooring, roofing and landscaping estimates.

Common mistakes

• Using a slanted leg as the height. The height is the perpendicular distance between the bases. Measuring along the leg gives a longer number and inflates the area.
• Forgetting to halve. ½(a + b)h is not (a + b)h. The factor of one-half is the whole reason two trapezoids make one parallelogram.
• Averaging all four sides. The area is not the average of the sides times the height. Only the two parallel sides go into the average.
• Mixing units. A base in feet and a height in inches will produce nonsense. Convert everything to one unit first — this calculator's length-unit selector applies to every field at once.
• Treating the trapezoid and trapezium definitions as universal. In the US, a trapezium has no parallel sides. In the UK, a trapezium is what Americans call a trapezoid. Say which convention you mean if the answer matters.
• Assuming four sides always work. Four sides only determine a trapezoid if the bases are unequal and the implied height is real. If you pick bases that are too short relative to the legs, no trapezoid exists and the calculator will say so.
• Confusing area with perimeter. Perimeter is a length (cm), area is a surface (cm²). A shape can have a large perimeter and a small area, or the reverse.
• Using the exclusive definition to reject parallelograms. If your bases happen to be equal, the formula still works — it just becomes base × height. There is no need to treat it as an error.
• Mismeasuring the perpendicular height on a slanted wall. On a slanted ceiling or roof, the perpendicular distance is not the vertical rise — it is the shortest distance between the two parallel lines containing the bases. Use a plumb line or a laser measure with a perpendicular mode.

Trapezoid vs trapezium vs parallelogram vs kite

The kite is worth noting because its area formula, ½ d₁ d₂, looks similar to the trapezoid's diagonal formula — but the kite's diagonals are perpendicular, so the sine term disappears (sin 90° = 1). The trapezoid's diagonal formula keeps the sine because the diagonals are not generally perpendicular.

Privacy

Every calculation, drawing, animation and conversion runs in your browser with plain JavaScript. Nothing you type is uploaded, logged or sent to a server. There is no account and no network request after the page loads. Recent calculations are optionally kept in your browser's localStorage so you can reload them, and clearing them removes them permanently from your device.

Limitations

• Four-sides mode needs to know which pair of sides is parallel. If you are not sure, the coordinates mode is a safer route.
• Four-sides mode requires the two bases to be different lengths; a parallelogram's height is not determined by its four sides alone.
• Coordinates mode assumes the four points are entered in order around the shape. Entering them in a crossed order will produce a self-intersecting polygon and a misleading area.
• Diagonals mode fixes the area but not the shape — infinitely many quadrilaterals share the same two diagonals and angle. The diagram is illustrative.
• Angles are entered and displayed in degrees, not radians.
• Extremely large or extremely small numbers may lose precision in the last digits; the calculator formats results to a sensible number of significant figures rather than pretending to unlimited accuracy.
• The interactive drag handles are available in Bases & Height mode. The other modes compute the shape from their inputs and are read-only by design.
• Snap-to-grid uses the current zoom level; when zoomed out, the snap step is coarser. Zoom in for fine adjustment.

Frequently asked questions

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