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Trapezoid Calculator

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Range: 0 cm – 1,000,000,000 cm

Result

28.0000 cm²

Area

Perimeter
24.0000 cm
Height
4.0000 cm

A trapezoid calculator takes the four sides of a trapezoid — two parallel bases and two legs — and returns the area, the perimeter and the height. A trapezoid is a four-sided shape with exactly one pair of parallel sides, which are called the bases; the other two sides are the legs, and they are what makes this shape harder than the rest of the pages in this subcategory. A rectangle, a parallelogram and a triangle all take a height directly, because the height is a length you can measure or draw. A trapezoid described by its four edges has no height in the list, and the height is exactly what the area formula needs — the average of the two bases times the height. So this page does the intermediate step for you: the two bases are parallel, so the difference between them, together with the two legs, forms a triangle, and the height of the trapezoid is the height of that triangle. A trapezoid with bases of 10 and 4 and legs of 5 and 5 works out as follows: the bases differ by 6, the legs are equal so each one claims 3 of that 6, and the triangle with a hypotenuse of 5 and a base of 3 has a height of 4. The area is then the average of the bases, 7, times 4, which is 28 square centimetres, and the perimeter is 4 + 10 + 5 + 5 = 24. One of the three readings comes out with almost no work at all: the perimeter is the four sides added up, and it never involves the height. The reason the page is worth having rather than doing by hand is the same triangle inequality that governs any triangle: the difference between the two bases has to be something the two legs can span. If the bases differ by 6 and the legs are 3 and 3, the legs can only just reach and the shape is a triangle with a doubled edge, a real answer with a height of 0. If the legs are 2 and 3, they cannot span 6 at all and no such trapezoid exists, so the page reports that rather than returning a negative number from the square root. The table below has one of those cases as a row: the seventh is a trapezoid whose legs add up to exactly the base difference — legs of 4 and 2 under bases of 10 and 4 — so it is folded flat and its area is 0. The other case cannot be a row, because legs that cannot span the difference are not a shape at all; that is what the page reports instead, and it is the reason a trapezoid's four sides are a stricter specification than they look. One more degenerate case is worth knowing before you start. If the two bases are equal, the shape is a parallelogram, and four sides no longer determine it: a parallelogram with sides 5, 5, 6 and 6 can be a rectangle 6 by 5, or leaned almost flat, and every one of those shapes has the same four sides and a different height. The page refuses that case rather than picking a height for you. If what you have is a parallelogram, use the parallelogram area calculator, which takes a base and a height; if you have the two bases and the height but not the legs, the area of a trapezoid calculator takes exactly that.

Trapezoids by their four sides, with the height and area each one gives

Top base (cm)Bottom base (cm)Left leg (cm)Right leg (cm)Height (cm)Area (cm²)
41055428
71054434
4745422
010684.824
01013131260
59554.582632.078
4104200
212774.89934.2929

Eight trapezoids, and the height column is the one the page exists to fill in — it is not a side, it is not printed on any drawing that gives four edges, and every other column here is an input. The first row is the one the page loads with, and it is the 3-4-5 triangle in disguise: equal legs of 5 over a base difference of 6 put a height of 4 in the middle. The second row is the same height reached with unequal legs of 5 and 4, which is what a scalene trapezoid looks like — the height moves off centre and the area changes even though the height does not. The fourth row is a triangle in a trapezoid's clothing, with a top base of 0 and the 6-8-10 right triangle for legs, and the fifth is another triangle of the same kind — top base 0 again, but this time the legs are 13 and 13 over a bottom base of 10, which is the isosceles 10-13-13 with a height of 12 and an area of 60. The seventh row is the edge of what the page accepts: legs of 4 and 2 add up to exactly the base difference of 6, so the height collapses to 0 and the area to 0, and the page answers rather than refusing. Shorter legs than those and there is no trapezoid at all. Every cell is recomputed from its row when the page is built, in centimetres and square centimetres.

Formula

A = ((a + b) ÷ 2) × h P = a + b + c + d x = (c² − d² + Δ²) ÷ (2Δ) h = √(c² − x²)

a, b
The two parallel sides, in centimetres — the bases. Either can be the longer one; the page works from the difference between them and does not care which you call the top. Their average, (a + b) ÷ 2, multiplied by the height, is the area, which is the formula the whole page is built around
c, d
The two legs, in centimetres: the sides that are not parallel. Their lengths decide whether the shape exists at all, because together they have to span the difference between the bases. They do not have to be equal — a trapezoid with legs of 5 and 4 is perfectly ordinary — and when they are equal the shape is called isosceles and the height is simply the square root of the leg squared minus half the base difference squared
Δ
The difference between the two bases, |b − a|, in centimetres. It is the base of the triangle that the two legs and the two bases enclose, and it is the quantity that decides which of the three cases you are in: smaller than the legs can span, exactly at the limit, or larger than they can reach. When it is 0 — equal bases — the shape is a parallelogram and the four sides stop determining the height, which is why the page reports an error rather than answering
x
Where the height lands along the base difference, in centimetres: the distance from one end of the difference to the foot of the perpendicular. It comes from the law of cosines applied to the triangle with sides c, d and Δ, and the page uses it only to get to the height. It can be outside the range from 0 to Δ, which is what happens when one leg leans outwards past the end of the shorter base — the shape is then still a trapezoid and the arithmetic still works
h
The height, in centimetres: the perpendicular distance between the two parallel bases, and the number the area formula needs. It comes out equal to the square root of c² − x², and it is printed to four decimals. Because it is a square root of a difference, the case where the legs cannot span the base difference is the case where the number under the root would go negative — which is why the page checks the triangle inequality first rather than letting the square root produce a nonsense value
A, P
The area in square centimetres and the perimeter in centimetres. The area is the average of the bases times the height, which is the same as half their sum times the height — a trapezoid and a rectangle of the same height cover the same ground when the rectangle's width is the average of the two bases. The perimeter is all four sides added, and it never involves the height at all

Use this page when the four edges are what you have. That is the usual case for land: a plot with two parallel sides and two that are not, measured around the boundary with a tape, where the parallel sides and the legs are all easy to measure and the perpendicular height is not — you would have to pace it out across the plot, which on a slope or through a crop is exactly what you cannot do. It is also the case for a panel cut to a drawing that gives four edges and no height, and for checking a shape somebody else has specified: enter the four sides and see whether they describe a trapezoid at all. The seventh row of the table is the one to look at for that check — legs of 4 and 2 with bases of 10 and 4 give a height of 0, because the difference of 6 is exactly what the two legs add up to and the shape has folded flat. Push either leg shorter and the page reports an error instead. Use the area of a trapezoid calculator when you have the two bases and the height and not the legs, since it takes that pair directly and skips the intermediate step. Use the parallelogram area calculator when the two bases are equal, because then the shape is a parallelogram and this page's four sides stop determining it. And use the triangle area calculator when the shorter base is 0 — a triangle is a trapezoid with one base of zero, and the fourth row of the table below is the 6-8-10 right triangle found that way, with an area of 24.

Worked examples

  1. Bases of 4 and 10 with legs of 5 and 5

    1. Difference between the bases: 10 − 4 = 6
    2. The legs are equal, so each claims half of it: 6 ÷ 2 = 3
    3. Height: √(5² − 3²) = √(25 − 9) = √16 = 4
    4. Average of the bases: (4 + 10) ÷ 2 = 7
    5. Area: 7 × 4 = 28
    6. Perimeter: 4 + 10 + 5 + 5 = 24

    The row the page loads with, and the one to check by hand, because the numbers are the 3-4-5 triangle in disguise: a leg of 5 and half the base difference of 3 give a height of 4 exactly. The height is also the one quantity you cannot read off the four sides, which is the whole point of the page. Compare the area with the rectangle that has the same height and a width of 7 — the average of the bases — and you get the same 28 square centimetres, which is the neatest way to see why the formula halves the sum rather than doing anything cleverer.

  2. Bases of 7 and 10 with legs of 5 and 4

    1. Difference between the bases: 10 − 7 = 3
    2. The legs are unequal, so the difference is split unevenly: (25 − 16 + 9) ÷ (2 × 3) = 18 ÷ 6 = 3
    3. Height: √(25 − 9) = √16 = 4
    4. Average of the bases: (7 + 10) ÷ 2 = 8.5
    5. Area: 8.5 × 4 = 34
    6. Perimeter: 7 + 10 + 5 + 4 = 26

    A scalene trapezoid, where the two legs are different lengths and the height lands somewhere other than the middle of the base difference. The answer here is that the 4 cm leg stands vertically between the right-hand ends of the two bases — the distance x comes out equal to the whole difference of 3, so the other leg takes none of it — which is what an unequal pair of legs does; the height then comes out equal to that leg's own length. The height still comes out at exactly 4, so this row and the first one have the same height and different areas, at 28 and 34 square centimetres, purely because their bases differ. That is a reminder that a trapezoid's four sides are all doing work: the same height of 4 sits on two different pairs of bases here, so it is the four of them together, not the height on its own, that fixes the area.

  3. Bases of 0 and 10 with legs of 6 and 8

    1. Difference between the bases: 10 − 0 = 10
    2. Unequal legs: (36 − 64 + 100) ÷ (2 × 10) = 72 ÷ 20 = 3.6
    3. Height: √(36 − 12.96) = √23.04 = 4.8
    4. Average of the bases: (0 + 10) ÷ 2 = 5
    5. Area: 5 × 4.8 = 24
    6. Perimeter: 0 + 10 + 6 + 8 = 24

    A triangle wearing a trapezoid's clothes: with one base of 0 the shape closes up into a triangle, and the numbers here are the 6-8-10 right triangle, whose area of 24 square centimetres is the same answer the triangle area calculator gives. The arithmetic is a reminder that the trapezoid formula contains the triangle formula inside it — with one base at zero, the average of the bases becomes half the other base, which is exactly the triangle's factor. The perimeter of a shape with a zero-length side is just the other three added, which here is also 24, a coincidence of these numbers rather than a rule.

  4. Bases of 4 and 10 with legs of 4 and 2

    1. Difference between the bases: 10 − 4 = 6
    2. The two legs add up to 4 + 2 = 6, exactly the difference
    3. Height: the legs can only just span the difference, so the height is 0
    4. Average of the bases: (4 + 10) ÷ 2 = 7
    5. Area: 7 × 0 = 0
    6. Perimeter: 4 + 10 + 4 + 2 = 20

    The edge of what the page accepts, and the reason it checks the legs before it takes a square root. Here the two legs add up to exactly the difference between the bases, so both of them lie flat along the longer base, the two bases are on top of one another and the shape encloses nothing. The page answers 0 rather than refusing, because those four sides really do describe a shape — a flat one. Shorten either leg by a centimetre and there is no such trapezoid at all, and that is the case reported as an error, on the grounds that a negative number under a square root is not a height.

Limitations

This page computes the area, the perimeter and the height of a trapezoid from its four sides, and nothing else. It does not give the angles, the diagonals, the length of the mid-segment or the position of the height along the bases, all of which the four sides do determine — the page simply does not report them, and the diagonal of a trapezoid in particular takes a second application of the same triangle arithmetic this page already does internally. It does not work backwards: there is no route from an area to the sides that produce it, and a trapezoid with a given area has infinitely many four-side descriptions. The outputs are in centimetres and square centimetres whatever units the inputs were entered in, since the dropdowns convert each side to centimetres before the arithmetic. The page assumes the shape is a trapezoid with exactly one pair of parallel sides and that the two sides you enter as bases are the parallel pair — swap a base and a leg and the page will compute the height of a different shape and report no error, because four numbers that form a valid triangle with the base difference always produce an answer. Both bases are treated as lengths that cannot be negative. Equal bases are reported as an error rather than answered, which is deliberate but is worth knowing: the shape is a parallelogram in that case and its height is genuinely not determined by its four sides, so a page that returned a number would be inventing one. The page does not say which of the three degenerate cases you are in beyond the error message: a folded trapezoid with a height of 0 is answered, while legs too short to span the base difference are refused, and the difference between the two is worth one centimetre on one leg. Four decimals is a display width shared with the other area pages in this subcategory, not a claim about precision — a side measured to the nearest centimetre moves the height by more than the fourth decimal place. Nothing here accounts for waste, kerf or uneven ground, so the area returned is the area of the shape as described rather than the quantity of material to order.

Frequently asked questions

How do I find the area of a trapezoid from its four sides?
Find the height first, then average the bases and multiply. The two bases differ by some amount Δ; that difference, together with the two legs, forms a triangle, and the height of the trapezoid is the height of that triangle. For bases of 4 and 10 with legs of 5 and 5: the difference is 6, each leg claims 3 of it, the height is √(25 − 9) = 4, the average of the bases is 7, and the area is 28 square centimetres.
Why does the page refuse some sets of four sides?
Because most sets of four sides do not describe a trapezoid. The two legs have to be able to span the difference between the bases, exactly as the two shorter sides of a triangle have to be able to span the longest one. Bases of 10 and 4 differ by 6, so legs of 4 and 2 just barely reach — the height is 0 — while legs of 3 and 2 cannot span 6 at all, and no such shape exists. The page reports that instead of returning a negative number from the square root.
What if the two bases are the same length?
Then the shape is a parallelogram, and its four sides no longer determine it. Sides of 5, 5, 6 and 6 describe a rectangle 6 by 5 with a height of 5, and also a shape leaned almost flat with a height of nearly nothing — same four sides, different areas. This page reports an error rather than choosing one for you. The parallelogram area calculator takes a base and a height instead and will answer that question directly.
What is an isosceles trapezoid and how does it simplify things?
One whose two legs are equal. The base difference then splits evenly between the two legs, so each leg and half the difference form a right-angled triangle, and the height is √(leg² − (Δ ÷ 2)²). With bases of 4 and 10 and equal legs, Δ is 6, half of it is 3, and a leg of 5 gives √(25 − 9) = 4. The table's first row is this case, and its shape is symmetric — the two base angles are equal and so are the two diagonals.
Is a triangle a special case of a trapezoid?
Arithmetically yes, when one of the bases is 0. The average of the bases then becomes half the remaining base, and the area formula turns into half a base times the height, which is the triangle formula. Enter a top base of 0 with legs of 6 and 8 and a bottom base of 10 and the page returns 24 square centimetres, the same answer the triangle area calculator gives for a 6-8-10 triangle. Whether a triangle counts as a trapezoid is a matter of definition, but the formula does not mind.
Why is the height not one of the four sides?
Because in a trapezoid it is not a side at all — it is the perpendicular distance between the two parallel bases, and inside the shape rather than on its boundary. It is the one quantity you cannot get by running a tape around the edge, which is why a page like this one is useful: it derives the height from the four sides you can measure. The height equals the shorter leg only in the degenerate case where that leg stands perpendicular to both bases.

References

Related calculators

Area of a Trapezoid CalculatorArea of a trapezoid from its two bases and height.Parallelogram Area CalculatorArea and perimeter of a parallelogram from its base, height and sloping side.Triangle Area CalculatorHalf a base times the perpendicular height above it.Rectangle CalculatorRectangle area, perimeter and diagonal from length and width.