Library / Geometry, coordinates, surveying and trigonometry / GEO-01
Geometry, coordinates, surveying and trigonometry
Parallelogram area
Area of a parallelogram from a chosen base and the perpendicular height to it.
- Ref
- GEO-01
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- A [cm²]
The area of a parallelogram is base times perpendicular height. Cutting off a triangle from one slanted end and moving it to the other end turns the parallelogram into a rectangle with the same base and height. Use the perpendicular height, not the length of the slanted side.
Area of a parallelogram from a chosen base and the perpendicular height to it. A sloping side can serve as the height only if it is perpendicular to the base.
Calculator
- Perimeter if the slanted side is 5 cm26 cm
- Area of the triangle with the same base and height, bh/212 cm²
- Height if the area is doubled6 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.
- A parallelogram with base 8 cm and height 3 cm has area 24 cm², the same as a rectangle 8 cm by 3 cm.
- Shearing the shape sideways changes the slanted sides but not the area, as long as the base and the height stay the same.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| h Perpendicular height | 0.5 to 20 cm | 4 | 160 | 156 | +10% | +10 % |
| b Base | 1 to 30 cm | 3 | 90 | 87 | +10% | +10 % |
Sweep of Base
| Base (cm) | Area (cm²) |
|---|---|
| 1 | 3 |
| 4.625 | 13.88 |
| 8.25 | 24.75 |
| 11.88 | 35.63 |
| 15.5 | 46.5 |
| 19.13 | 57.38 |
| 22.75 | 68.25 |
| 26.38 | 79.13 |
| 30 | 90 |
Sweep one variable
Variables and how each one contributes
Area
What the equation gives you. Everything else feeds it.
Base
The chosen base, in cm. A longer base means a larger area in direct proportion. Raising it by 10 % (to 8.8) moves the result by +10 %, from 24 to 26.4 cm².
Perpendicular height
The perpendicular distance between the base and the opposite side, in cm. A greater height gives a larger area in direct proportion. Raising it by 10 % (to 3.3) moves the result by +10 %, from 24 to 26.4 cm².
Worked example with the values above
- Choose base b = 8 cm and the perpendicular height h = 3 cm.
- Multiply: 8 × 3.
- A = 24 cm².
Assumptions and limits
- A flat (plane) shape with straight sides, measured in one consistent unit.
- For regular polygons all sides and all angles are equal.
Common mistakes
- Using a sloping side as the height when it is not perpendicular to the base.
- Mixing units, such as a base in mm and a height in cm.
- Using the diagonal of one shape in the formula for another.
Statistics and uncertainty
These are exact formulas, so the result is only as good as the lengths measured. Area scales with the square of a length, so a 1 % error in a length gives about a 2 % error in the area.
Where it is used
- Plate, panel and gasket areas from simple drawings.
- Checking the size of regular flanges, hex bars and tiles.
Maths you need
- Area: the flat space inside a shape, in square units
- Perpendicular: at a right angle to
- Tangent of an angle: opposite over adjacent