Library / Foundations: Maths / FMA-07
Foundations: Maths
Area of a trapezium
Area from the two parallel sides and the height between them.
- Ref
- FMA-07
- Rev
- v1
- Field
- Foundations: Maths
- Result
- A [cm²]
a and b are the parallel sides and h the perpendicular height between them. A triangle is the case a = 0, with area ½bh.
Average the two parallel sides, then multiply by the height between them.
Calculator
- Triangle area ½bh18 cm²
- Mean width7 cm
What drives the result
Each variable is moved across its typical range while the others stay at the values above. The longest bar is the key driver.
- The height is perpendicular to the parallel sides, not the slanted edge.
- The same average-width idea gives the area of a trapezoidal channel or ramp cross-section.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Height | 1 to 20 cm | 7 | 140 | 133 | +10% | +10 % |
| a Parallel side a | 1 to 30 cm | 20 | 78 | 58 | +10% | +3.57 % |
| b Parallel side b | 1 to 30 cm | 12 | 70 | 58 | +10% | +6.43 % |
Sweep of Height
| Height (cm) | Area (cm²) |
|---|---|
| 1 | 7 |
| 1.454 | 10.18 |
| 2.115 | 14.8 |
| 3.075 | 21.53 |
| 4.472 | 31.3 |
| 6.503 | 45.52 |
| 9.457 | 66.2 |
| 13.75 | 96.27 |
| 20 | 140 |
Sweep one variable
Variables and how each one contributes
Area
What the equation gives you. Everything else feeds it.
Parallel side a
One of the parallel sides. Directly raises the average width.
Parallel side b
The other parallel side. Directly raises the average width.
Height
Perpendicular distance between the parallel sides. Directly proportional.
Worked example with the values above
- Mean width: (5 + 9)/2 = 7 cm.
- Multiply by height 4 cm.
- A = 28 cm².
Assumptions and limits
- The two sides a and b are parallel.
- All lengths in the same unit.
Common mistakes
- Using the slant side for the height.
- Forgetting the factor ½.
Statistics and uncertainty
Measurement errors of a few per cent in each dimension add roughly in quadrature on the area: δA/A ≈ √[(δh/h)² + (δ(a+b)/(a+b))²].
Where it is used
- Cross-sectional areas of channels and embankments.
- Land and layout areas.
Maths you need
- Area: square units
- Averages: mean of two lengths