About the Trapezoid Calculator
A trapezoid is a four-sided shape with one pair of parallel sides. Picture a rectangle whose top edge has been shortened and the corners pulled in — the bottom and top are still parallel, but they are no longer the same length.
That shape turns up constantly the moment you stop working with tidy rectangles. A field bounded by a road and a river that are not quite parallel in width. A gable wall under a single-pitch roof. A retaining wall, thicker at its base. The cross-section of a drainage channel or an embankment. A tapered garden bed.
The area formula for it is one of the most quietly elegant in geometry, and this calculator will also work backwards — recovering the height from the four side lengths, which is the measurement you can never reach with a tape when the shape is already built.
How to Use the Trapezoid Calculator
Parallel sides and height. The direct case. Enter both parallel sides and the perpendicular distance between them.
Parallel sides and both legs. When you can measure all four edges but not the height. The calculator recovers the height from them.
Isosceles. For the symmetrical case where both slanted legs are the same length — enter it once.
The parallel sides can go in either order; only the difference between them matters.
The Formula
Area: A = ½ (a + b) h
Median: m = (a + b) / 2 the average of the parallel sides
So: A = m × h
Perimeter: P = a + b + c + dWritten as A = m × h, the formula stops looking like something to memorise. The median is the average of the two parallel sides, and multiplying it by the height is exactly the rectangle rule.
That is the whole idea: a trapezoid has the same area as a rectangle of the same height whose width is the average of the top and bottom. Too wide at one end, too narrow at the other, and the two errors cancel precisely.
It also explains the boundary cases. Make both parallel sides equal and the average is that value — the formula collapses to the rectangle one. Shrink the top to nothing and the average is half the base, giving ½ × base × height, the triangle formula. The trapezoid formula contains both.
The median is a real line
The median is not just an average — it is the line joining the midpoints of the two slanted legs, sometimes called the mid-segment. Its length genuinely is the average of the two parallel sides, and it sits exactly halfway up. So you can measure it directly on site instead of measuring both parallel sides, which is occasionally the easier job.
Recovering the Height From the Four Sides
This is the part worth understanding, because it is where the arithmetic gets real.
The longer parallel side overhangs the shorter one, and the total overhang is b − a. That overhang splits between the two ends, and each end forms a right triangle whose vertical leg is the height:
x + y = b - a
c² - h² = x² (the end carrying leg c)
d² - h² = y² (the end carrying leg d)Eliminating y and h gives the split directly:
x = ( (b-a)² + c² - d² ) / ( 2(b-a) )
h = √( c² - x² )In the isosceles case the two ends are identical, so x is simply half the overhang and the formula reduces to h = √(c² − ((b−a)/2)²).
When four sides do not make a trapezoid
Two cases fail, and both are real geometry rather than arithmetic trouble.
The legs are too short to reach. Parallel sides of 2 and 20 leave an 18-unit overhang. Legs of 3 cannot span it — the shape simply does not close, and the square root above goes negative. The calculator says so rather than returning NaN.
Both parallel sides are equal. Then the figure is a parallelogram, and a parallelogram is not rigid. Push its top sideways and every side length stays identical while the height changes. Four equal-paired sides genuinely do not determine the height, so the calculator asks for the height directly instead of picking one of the infinitely many possibilities.
That second case is the same rigidity point that makes triangles structurally special: a triangle's three sides fix it completely, a quadrilateral's four do not.
Step-by-Step Example
With the height. Parallel sides of 8 and 14 metres, 5 metres apart.
m = (8 + 14) / 2 = 11
A = 11 × 5 = 55 m²Eleven metres is the width of the rectangle that would cover the same ground.
From the four sides. Parallel sides 8 and 12, legs 3 and 5.
overhang = 12 - 8 = 4
x = (4² + 3² - 5²) / (2 × 4) = (16 + 9 - 25) / 8 = 0
h = √(3² - 0²) = 3An x of zero means that leg is vertical — this is a right trapezoid, with one leg perpendicular to both parallel sides. The area is 10 × 3 = 30, and the perimeter is 28.
Isosceles. Parallel sides 8 and 14, both legs 5.
overhang = 6, so 3 at each end
h = √(5² - 3²) = √16 = 4
A = 11 × 4 = 44The 3-4-5 triangle appearing at each end is why these numbers come out whole.
Understanding Your Result
The area is in square units of whatever length unit you entered.
The height is the perpendicular gap between the parallel sides — given back to you when the calculator derived it from the legs.
The perimeter is all four sides. In height mode it is reported as not determined, and that is not a limitation to work around: the legs can lean at any angle without changing the height, so three numbers genuinely do not fix the perimeter.
The median is the average of the parallel sides, and the line halfway up.
The sides are listed together so you can check what the calculator worked from.
When Should You Use This Calculator?
Land and fields. A plot between two non-parallel boundaries is a trapezoid, and its area sets the price or the seed quantity.
Gable and skillion walls. A wall under a single-pitch roof is a trapezoid, which gives the cladding, plaster or paint area.
Retaining walls and embankments. Their cross-section is a trapezoid, and the area times the length gives the volume of material.
Drainage channels and ditches. A trapezoidal channel is standard, because the sloped sides resist collapse. Cross-sectional area drives the flow capacity.
Decking and patios. Tapered areas are trapezoids; work out the area before ordering boards.
Roof faces. A hip roof's long faces are trapezoids, and each needs its own area for tiles or felt.
Bridges and trusses. Trapezoidal panels are common, and their geometry sets the member lengths.
Table tops and tapered furniture. A trapezoid table seats people around a taper, and its area sets the material.
Common Mistakes
Using a slanted leg as the height. The most common error with this shape by some distance. The height is the perpendicular gap; a leg is longer, so the area always comes out too big. If you only have the legs, use the four-sides mode and let the calculator find the height properly.
Adding the parallel sides without halving. (a + b) × h is twice the area. The ½ is not optional.
Halving only one side. It is (a + b)/2, the average of both — not a/2 + b.
Using the two legs in the area formula. Only the parallel sides go into the average. The legs affect the perimeter and, indirectly, the height, but they never appear in the area formula.
Assuming symmetry. Most trapezoids are not isosceles. Unless you know both legs are equal, use the two-leg mode, because the overhang rarely splits evenly.
Expecting four sides to fix the shape. They do for a trapezoid with unequal parallel sides, but not for a parallelogram. Quadrilaterals are flexible in a way triangles are not.
Mixing up the names. In British usage this shape is a trapezium and "trapezoid" means a quadrilateral with no parallel sides at all. The formulas here apply to the shape with exactly one parallel pair, whichever word your textbook uses.