Trapezoid Area Calculator

The area of a trapezoid is A = ½(b₁ + b₂) × h: the average of the two parallel sides multiplied by the distance between them. Enter both bases and the height above and the calculator shows the working.

What the formula means

A trapezoid has exactly one pair of parallel sides, called the bases. They are usually different lengths, and the shape tapers between them. Averaging the two bases produces the width of an equivalent rectangle of the same height and the same area, which is why the formula looks like the rectangle one with an average substituted for the width.

Note the naming. In American usage a trapezoid has one parallel pair and a trapezium has none. In British usage the two words are swapped. The formula is the same either way, so the terminology matters only when reading a textbook from the other side of the Atlantic.

A worked example

Take bases of 30 and 45 units and a height of 20.

• Add the bases: 30 + 45 = 75
• Halve that for the average: 37.5
• Multiply by the height: 37.5 × 20 = 750 square units

The check is that the answer must sit between the two rectangles you could draw. A 30 by 20 rectangle covers 600 and a 45 by 20 covers 900, and 750 sits neatly between them, as it must.

Height means perpendicular height

The height is the shortest distance between the two parallel sides, measured at a right angle to both. The sloping sides are longer, and using one of them inflates the answer.

If you have measured a slanted side and know the horizontal offset, recover the true height with Pythagoras. A trapezoid whose slanted side is 13 units with a horizontal run of 5 has a height of √(169 − 25) = 12.

Why this shape appears everywhere in earthworks

Dig a trench or build an embankment and the sides cannot stand vertical in loose material, so they batter outward. The cross-section that results is a trapezoid, and its area multiplied by the length gives the volume of soil moved.

A drainage ditch 1 m wide at the bottom, 2.5 m wide at the top and 0.8 m deep has a cross-section of ½(1 + 2.5) × 0.8 = 1.4 m². Over 60 m that is 84 m³ of spoil, which is around six lorry loads at 14 m³ each.

The same shape governs canal sections, road embankments and the profile of a dam, and in each case the trapezoid area is the first step to a volume.

The trapezoidal rule

This formula does more work in numerical methods than it does in geometry. To find the area under a curve when there is no neat formula for it, divide the region into thin vertical strips. Each strip is close to a trapezoid, since the curve across a narrow strip is nearly straight.

Add the strip areas and you have an estimate of the integral. Halve the strip width and the error falls by about a factor of four. This is the trapezoidal rule, and it is how spreadsheets integrate measured data, how a flow meter turns a varying rate into a total, and how a GPS turns a speed trace into a distance.

Where else it turns up

Roofing and cladding. A hip roof face is a trapezoid, and panels are ordered by its area.

Land parcels. A plot bounded by two parallel roads and two side boundaries is a trapezoid, and its area comes straight from this formula.

Structural sections. Tapered beams and bridge girders have trapezoidal profiles, and section area feeds into weight and stiffness.

Depreciation and finance. A quantity falling steadily from one value to another over a period accumulates a trapezoidal total, which is why the average of the opening and closing figure multiplied by the term gives the right answer.

Finding a missing dimension

The formula rearranges when the area is known and one measurement is not. To find the height from a known area, h = 2A / (b₁ + b₂). A trapezoid covering 750 square units with bases of 30 and 45 has a height of 1500 / 75 = 20.

To find a missing base, b₁ = 2A/h − b₂. This comes up in land parcels where the area is on the deed and one boundary is obscured, and in earthworks where the cut volume is fixed by the specification and the top width has to follow from it.

Splitting a trapezoid into simpler shapes

If the formula slips your memory, drop a perpendicular from each end of the shorter base to the longer one. The trapezoid becomes a rectangle with a right triangle at each end.

For bases of 30 and 45 with a height of 20, the rectangle is 30 by 20 and covers 600. The two triangles together have a combined base of 45 − 30 = 15 and a height of 20, so they total ½ × 15 × 20 = 150. Adding gives 750, the same answer. This decomposition is also how you handle a trapezoid whose sloping sides differ, since the two triangles need not be equal.

Isosceles trapezoids and perimeter

An isosceles trapezoid has equal sloping sides and a line of symmetry. Its sloping side length is √(h² + ((b₂−b₁)/2)²). For bases of 30 and 45 with a height of 20, each slope is √(400 + 56.25) = 21.36, so the perimeter is 30 + 45 + 2 × 21.36 = 117.72.

Perimeter matters when the area does not: edging a raised bed, framing a panel or fencing a plot all depend on the distance round rather than the space inside.

Where the shape hides in plain sight

A great deal of everyday geometry is trapezoidal without being labelled that way. A bucket seen from the side, a lampshade in profile, the cross-section of a river channel, the side of a wheelbarrow, and the outline of a hip roof are all trapezoids.

The reason is that the shape is what you get whenever two parallel edges of different length are joined, which happens any time something tapers. Recognising it saves reaching for a more complicated method: measure the two parallel edges and the perpendicular distance between them, and the area follows.

Averaging as a general idea

The trapezoid formula is really a statement about averages: a quantity changing steadily from one value to another behaves, in total, as though it held the average of the two throughout.

That is why a vehicle accelerating evenly from 10 to 30 m/s over 8 seconds covers the same distance as one travelling steadily at 20 m/s for those 8 seconds, which is 160 m. Plot speed against time and the region under the line is a trapezoid with bases of 10 and 30 and a width of 8, giving ½(10 + 30) × 8 = 160.

The shape and the physics are the same calculation. Recognising that saves memorising a separate equation of motion, and it explains why so many problems involving steady change reduce to averaging the endpoints.

Common mistakes

Using a slanted side as the height. Only the perpendicular distance between the parallel sides belongs in the formula.

Adding the wrong pair. The two values averaged must be the parallel sides, not any two sides.

Forgetting the half. Adding the bases and multiplying by the height without halving doubles the answer.

Assuming the shape is symmetric. An isosceles trapezoid has equal sloping sides, but the formula does not require it and works for any trapezoid.