Calculator Features
🧭 Jump to a section
- What is a trapezoid
- Trapezoid vs trapezium
- Parts of a trapezoid
- Types of trapezoids
- Area formula
- Perimeter formula
- Height formula
- Finding a missing base
- Finding missing legs
- Median / midsegment
- Interior angles
- Diagonals
- Coordinate geometry method
- Worked examples
- Applications
- Real-world examples
- Glossary
- Common mistakes
- Practice questions
- Printable formula sheet
What Is a Trapezoid or Trapezium?
A trapezoid is a four-sided shape (a quadrilateral) with exactly one pair of parallel sides. Those parallel sides are called the bases (labelled a and b), and the two non-parallel sides connecting them are the legs. Unlike a rectangle or parallelogram, a trapezoid's legs generally aren't equal or parallel to each other — which is exactly what makes it flexible enough to model roof sections, drainage channels, and land parcels that a simple rectangle can't.
Trapezoid vs Trapezium: Same Shape, Different Word
This is one of the most confusing terminology splits in geometry, purely because the two English-speaking traditions swapped the words:
| Region | "Trapezoid" means | "Trapezium" means |
|---|---|---|
| United States, Canada | One pair of parallel sides (this shape) | No parallel sides at all |
| Australia, United Kingdom | No parallel sides at all | One pair of parallel sides (this shape) |
This calculator uses "trapezoid" in its name and URL for broad international search reach, but the shape it solves — one pair of parallel sides — is exactly what Australian and UK readers know as a trapezium. Wherever "trapezoid" appears on this page, Australian and UK readers can substitute "trapezium" with no change in meaning.
Parts of a Trapezoid
- Bases (a, b) — the two parallel sides. By convention, a is often the longer base.
- Legs (c, d) — the two non-parallel sides connecting the bases.
- Height (h) — the perpendicular distance between the two bases.
- Median / midsegment (m) — the segment joining the midpoints of the two legs, parallel to both bases, with length equal to their average.
- Diagonals — the two segments connecting opposite corners.
Types of Trapezoids
General Trapezoid
No special symmetry — the two legs can be any length, and the base angles can all differ. A general trapezoid is fully determined by its four side lengths (both bases and both legs), because that's exactly enough information to fix the height and both legs' horizontal offsets — see the coordinate geometry method below for the derivation.
Isosceles Trapezoid
The two legs are equal in length, and the base angles on each side match. This symmetry means the top base is centred over the bottom one, so an isosceles trapezoid is fully determined by its two bases plus just one more value — height, leg length, or area.
Right Trapezoid
One leg is perpendicular to both bases, forming two right angles on that side. The perpendicular leg is simply equal to the height, which again means a right trapezoid is fully determined by its two bases plus just one more value.
Trapezoid Area Formula
Average the two parallel bases, then multiply by the height. This works for any trapezoid — general, isosceles or right — because it doesn't depend on the leg lengths or angles at all, only on the bases and the perpendicular height between them. A trapezoid with bases 10 and 6 and height 4 has an area of ((10+6)/2)×4 = 32 square units.
Perimeter Formula
Simply add all four sides. Unlike area, perimeter does depend on the leg lengths, which is why a right or isosceles assumption (or all four sides directly) is needed to calculate it — bases and height alone aren't enough.
Height Formula
The height can be found several ways depending on what's known:
| Known | Height formula |
|---|---|
| Area and both bases | h = 2A / (a + b) |
| Isosceles trapezoid, bases and leg | h = √(leg² − ((a−b)/2)²) |
| Right trapezoid, bases and slanted leg | h = √(d² − (a−b)²) |
| General trapezoid, all four sides | See the coordinate geometry method |
Finding a Missing Base
Rearranging the area formula: b = 2A/h − a (or swap a and b to solve for the other). This is the most common "missing base" scenario — you know the area and height (perhaps from a survey or a roof pitch) and need the width of the far edge.
Finding Missing Legs
For an isosceles trapezoid: leg = √(h² + ((a−b)/2)²). For a right trapezoid: the perpendicular leg equals h directly, and the slanted leg = √(h² + (a−b)²). For a general trapezoid, a single missing leg can't be found from the bases and height alone — the other leg's length is also needed, because infinitely many trapezoids share the same bases and height with different leg pairs.
Median (Midsegment)
The median is the same length as the average of the two bases — and, conveniently, area can also be written as A = m × h, since the median is exactly the "average width" the height formula is implicitly using.
Interior Angles
A trapezoid's four interior angles always sum to 360°, like any quadrilateral. Because the two bases are parallel, the angles on each leg are supplementary — the angle at the bottom of a leg plus the angle at the top of that same leg always equals 180° (co-interior angles on a transversal cutting two parallel lines). In a right trapezoid, one leg contributes two 90° angles. In an isosceles trapezoid, both base angles on the longer base are equal, and both base angles on the shorter base are equal.
Diagonals
A trapezoid's two diagonals are generally different lengths, except in the isosceles case, where they're always equal: p = q = √(ab + c²). This is a useful practical check — if you measure a trapezoid's diagonals and they're equal, the shape is isosceles (or a rectangle, which is a special case of both isosceles and right).
Coordinate Geometry Method (General Trapezoid)
This calculator solves a general trapezoid from its four sides by placing it on a coordinate plane: base a runs from (0, 0) to (a, 0), and base b sits parallel at height h, from (e₁, h) to (e₁+b, h), where e₁ is the left leg's horizontal offset. The right leg's offset e₂ satisfies e₁ + e₂ = a − b — a relationship fixed by the bases alone, before the legs are even considered. From there, c² = e₁² + h² and d² = e₂² + h² give two equations in two unknowns (e₁ and h), which solve directly to:
h = √(c² − e₁²)
If the result under the square root is negative, the four side lengths entered cannot form a real trapezoid — a specific, checkable form of "impossible input" this calculator always tests for before showing a result.
Worked Examples
Area from two bases and height
Missing height from area and bases
Missing base from area, height and one base
Isosceles trapezoid legs and diagonals
Right trapezoid example
Roofing example — a lean-to roof section
Land-surveying example — an isosceles boundary
Impossible-input example
Construction, Roofing and Surveying Applications
| Field | Use |
|---|---|
| Roofing | Lean-to and skillion roof panels, hip-roof end sections, gutter cross-sections |
| Construction | Retaining wall cross-sections, tapered footings, stair stringers |
| Land surveying | Irregular boundary parcels with two roughly parallel sides |
| Civil engineering | Drainage channel and canal cross-sections, road batters |
| Architecture | Tapered window and door openings, lampshade and planter panels |
| Manufacturing | Sheet-metal transition pieces, conveyor hoppers |
Real-World Examples
Lean-to and skillion panels
Tapered pier & abutment sections
Canal & culvert cross-sections
Boundary parcels, two parallel edges
Tapered openings, planter panels
Hoppers, transition pieces
Glossary
- Base
- One of the two parallel sides of a trapezoid.
- Leg
- One of the two non-parallel sides connecting the bases.
- Height
- The perpendicular distance between the two bases.
- Median (midsegment)
- The segment joining the midpoints of the legs; length equals the average of the two bases.
- Isosceles trapezoid
- A trapezoid whose two legs are equal in length.
- Right trapezoid
- A trapezoid with two right angles on one leg.
- Diagonal
- A line segment connecting two opposite (non-adjacent) corners.
- Centroid
- The trapezoid's geometric centre of area.
- Trapezium (AU/UK)
- The Australian and British word for this exact shape — see the terminology section above.
Common Mistakes
| Mistake | Fix |
|---|---|
| Using a leg length as the height | The height is the perpendicular distance between the bases, not a slanted leg's length |
| Assuming bases and height alone fix the legs | They don't, unless the trapezoid is isosceles or right — a general trapezoid needs all four sides |
| Forgetting the ÷2 in the area formula | A = ((a+b)/2)×h, not (a+b)×h |
| Mixing up "trapezoid" and "trapezium" across regions | Check which country's convention applies — see the terminology section |
| Assuming all four angles are equal | Only the co-interior pairs on each leg are supplementary, not all four angles |
| Treating a parallelogram as a trapezoid input | If both bases are equal, it's a parallelogram — a trapezoid's height isn't determined by side lengths alone in that case |
Formula Cheat Sheet
Quick Reference
Area:A = ((a+b)/2)×h | Perimeter: P = a+b+c+dMedian:
m = (a+b)/2 | Height from area: h = 2A/(a+b)Isosceles leg:
c = d = √(h² + ((a−b)/2)²)Isosceles diagonal:
p = q = √(ab + c²)Right trapezoid:
perpendicular leg = h, slanted leg = √(h² + (a−b)²)Interior angles: co-interior pairs sum to 180°; all four sum to 360°
Practice Questions
Beginner (with answers)
- Find the area of a trapezoid with bases 8 and 5, height 4.
- Find the median of a trapezoid with bases 12 and 7.
- An isosceles trapezoid has bases 9 and 5, height 3. Find each leg.
- A right trapezoid has bases 7 and 4, height 5. Find the slanted leg.
- Find the perimeter of a trapezoid with sides 8, 5, 4 and 4.5.
Show answers
1) ((8+5)/2)×4=26 2) (12+7)/2=9.5 3) √(9+4)=√13≈3.606 4) √(25+9)=√34≈5.831 5) 8+5+4+4.5=21.5
Advanced (with answers)
- A trapezoid has area 45, bases 12 and 6. Find the height.
- A trapezoid has area 40, height 5, base a = 9. Find base b.
- An isosceles trapezoid has bases 10 and 4, leg 5. Find the height and area.
- Find the diagonal of an isosceles trapezoid with bases 8 and 4, leg 5.
- A general trapezoid has bases 12 and 8, legs 5 and 6. Find its height (2 dp).
Show answers
1) h=2×45/18=5 2) b=2×40/5−9=7 3) half=3, h=√(25−9)=4, A=((10+4)/2)×4=28 4) √(8×4+25)=√57≈7.55 5) S=4, diff=(25−36)/4=−2.75, e1=0.625, h=√(25−0.390625)≈4.96
🔑 Key Takeaways
- A trapezoid has exactly one pair of parallel sides (bases); Australia and the UK call this shape a trapezium.
- Area A = ((a+b)/2)×h works for every trapezoid, regardless of leg shape.
- A general trapezoid needs all four sides to be uniquely solved; isosceles and right trapezoids need only the bases plus one more value.
- Isosceles trapezoids have equal legs and equal diagonals; right trapezoids have one leg equal to the height.
- Bases and height alone never determine a general trapezoid's legs or angles — the top base could sit anywhere along its length.
Frequently Asked Questions
What is a trapezoid?
A four-sided shape with exactly one pair of parallel sides, called the bases. The other two sides, called legs, are generally not parallel to each other.
What is a trapezium?
In Australia and the UK, "trapezium" means exactly the same shape as "trapezoid" in the US — one pair of parallel sides. Confusingly, in the US "trapezium" means a quadrilateral with no parallel sides at all, the reverse of the Australian/UK meaning.
Is a trapezoid the same as a trapezium?
For Australian and UK readers, yes — "trapezium" is your word for exactly this shape. For US readers, "trapezoid" is the matching term; "trapezium" means something different (no parallel sides) in American usage.
How do you calculate the area of a trapezoid?
A = ((a + b) / 2) × h — average the two parallel bases, then multiply by the height. Bases 10 and 6 with height 4 give an area of 32 square units.
How do you calculate the area of a trapezium?
The same formula as a trapezoid: A = ((a + b) / 2) × h, since "trapezium" (Australia/UK) and "trapezoid" (US) name the same shape.
How do you calculate the perimeter of a trapezoid?
Add all four sides: P = a + b + c + d. Unlike area, this needs the actual leg lengths, not just the bases and height.
How do you find the height of a trapezoid?
From area and both bases: h = 2A / (a+b). From an isosceles leg: h = √(leg² − ((a−b)/2)²). From a right trapezoid's slanted leg: h = √(d² − (a−b)²).
How do you find a missing base of a trapezoid?
Rearrange the area formula: b = 2A/h − a (or swap a and b to solve for the other base), given the area, height and one base.
How do you find a missing leg of a trapezoid?
For an isosceles trapezoid: leg = √(h² + ((a−b)/2)²). For a right trapezoid: the perpendicular leg equals h, and the slanted leg = √(h² + (a−b)²). A general trapezoid's single missing leg can't be found from bases and height alone — the other leg is also needed.
What is the median or midsegment of a trapezoid?
The segment joining the midpoints of the two legs, parallel to both bases, with length m = (a+b)/2 — the average of the two bases.
What is an isosceles trapezoid?
A trapezoid whose two legs are equal in length, making it symmetric about a vertical line through the midpoints of both bases. Its diagonals are also always equal.
What is a right trapezoid?
A trapezoid with two right angles on one side, meaning one leg is perpendicular to both bases. That perpendicular leg is always equal to the height.
How do you calculate the diagonals of a trapezoid?
For an isosceles trapezoid, both diagonals equal √(ab + c²), where c is the leg length. For a general trapezoid, the two diagonals differ and are found from the trapezoid's coordinates.
How do you calculate the angles of a trapezoid?
The four interior angles always sum to 360°. Because the bases are parallel, each leg's two angles (top and bottom) are supplementary, summing to 180°.
Can a trapezoid have four different side lengths?
Yes — that's a general (scalene) trapezoid. It's still fully solvable given all four sides, using coordinate geometry to find the height.
Can I calculate a trapezoid from just the bases and height?
You can find the area, but not the individual leg lengths or angles — bases and height alone don't uniquely determine a general trapezoid, since the top base could sit at many different horizontal positions and still give the same area. Choose the isosceles or right type if that assumption applies.
Why can't four side lengths always make a trapezoid?
The two legs must be long enough to span the horizontal gap created by the difference in base lengths. If they're too short, no real height exists, and the calculator reports the combination as geometrically impossible rather than guessing.
What if both bases are the same length?
Then it's a parallelogram, not a trapezoid — its height isn't determined by the side lengths alone, since a parallelogram can lean at any angle while keeping the same side lengths.
What is the centroid of a trapezoid?
The geometric centre of area, located along the line joining the midpoints of the bases, at a distance of h(2b+a)/(3(a+b)) from base a.
What units are used for trapezoids?
Any linear unit for the bases, legs, height, median and diagonals (mm, cm, m, ft); area uses the squared version of the same unit (m², ft²).
Can I calculate a trapezoid online?
Yes — choose General, Isosceles or Right above, enter the required values, and the calculator solves everything else instantly with the working shown.
How accurate is a trapezoid calculator?
This calculator uses standard double-precision arithmetic, but displayed results are rounded and cannot be more accurate than the measurements entered.
Can students use this calculator?
Yes — enter the known values for whichever type matches, and the calculator shows the formula and full working, useful for checking homework as well as producing an answer.
What are common trapezoid mistakes?
Using a slanted leg's length as the height; assuming bases and height alone fix the legs; forgetting the ÷2 in the area formula; and mixing up regional "trapezoid"/"trapezium" terminology.
How is a trapezoid used in roofing?
Lean-to and skillion roof sections are often right trapezoids, with the eave-to-ridge rise as the height and the rafter as the slanted leg — directly giving both roofing area and rafter length.
How is a trapezoid used in construction?
Tapered footings, retaining wall cross-sections and stair stringers are commonly trapezoidal, with the area formula giving material volume per linear metre.
How is a trapezoid used in surveying?
Land parcels with two roughly parallel boundaries are modelled as trapezoids, with the area formula giving the parcel area directly from a frontage, rear boundary and depth measurement.
What happens if I enter more values than needed?
The calculator uses the required values to solve the shape and cross-checks any extra values you entered (such as perimeter or area) against the result, flagging a warning if they don't match.
Can I export the results?
Yes — use Copy Results to copy everything to the clipboard, Copy Formula for just the formulas used, Export CSV for a spreadsheet-ready file, or Print / Save as PDF for a printable worksheet.
Does a rectangle count as a trapezoid?
Under the "at least one pair of parallel sides" definition used in most modern curricula (including this calculator), yes — a rectangle is a special trapezoid where both pairs of sides are parallel. This calculator will classify a = b with right angles as a rectangle explicitly.
What's the difference between a trapezoid and a parallelogram?
A trapezoid has exactly one pair of parallel sides (or at least one, depending on definition); a parallelogram has two pairs. If a trapezoid's bases become equal in length, it becomes a parallelogram.
Why does the isosceles trapezoid formula only need the bases and one more value?
Because symmetry forces the top base to sit exactly centred over the bottom one — both legs' horizontal offsets are automatically (a−b)/2, removing the ambiguity a general trapezoid has.
Why does the right trapezoid formula only need the bases and one more value?
Because one leg is fixed perpendicular (offset zero), which pins down the shape just as completely as the isosceles symmetry does, leaving only the slanted leg free to be calculated.
Can a trapezoid have a right angle?
Yes — a right trapezoid has exactly two right angles, both on the same leg. It cannot have three or four right angles without becoming a rectangle.
Calculation Assumptions
ℹ️ What this calculator assumes
- Results are calculated using Euclidean (flat-plane) geometry.
- Measurements are assumed to lie in a flat plane, not on a curved or sloped surface.
- Rounding to the selected decimal precision may cause very small differences between displayed values and hand calculations.
- All input values are assumed to use the same unit — mixing units (e.g. a base in metres and a leg in centimetres) will produce an incorrect result.
About This Calculator
Printable Formula Sheet
A one-page reference with every trapezoid formula on this page.
Trapezoid (Trapezium) Formula Sheet
MegaCalcOnline.com · General, isosceles and right trapezoid formulas
Scan for the live calculator
| Find | Formula |
|---|---|
| Area | A = ((a+b)/2) × h |
| Perimeter | P = a + b + c + d |
| Median / midsegment | m = (a+b)/2 |
| Height from area | h = 2A / (a+b) |
| Isosceles leg | c = d = √(h² + ((a−b)/2)²) |
| Isosceles diagonal | p = q = √(ab + c²) |
| Right trapezoid perpendicular leg | = h |
| Right trapezoid slanted leg | √(h² + (a−b)²) |
| Sum of interior angles | 360° |
References
- Khan Academy — Quadrilaterals and trapezoids
- OpenStax Mathematics — geometry and measurement
- ACARA — Australian Curriculum: Mathematics
- NIST — SI units for length and area
- Encyclopaedia Britannica — Trapezoid
- Wolfram MathWorld — Trapezoid