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Geometry, coordinates, surveying and trigonometry
Kite area from its diagonals
Area of a convex kite from its two full diagonals, which cross at right angles.
- Ref
- GEO-02
- Rev
- v1
- Field
- Geometry, coordinates, surveying and trigonometry
- Result
- A [cm²]
A kite has two perpendicular diagonals. The longer one is cut at right angles by the shorter, and the four right triangles formed fill half of the rectangle d₁ × d₂ drawn round the diagonals. So the area is d₁d₂/2. The same rule holds for a rhombus and any quadrilateral with perpendicular diagonals.
Area of a convex kite from its two full diagonals, which cross at right angles. The four right triangles fill half of the rectangle drawn round the diagonals.
Calculator
- Area of the rectangle round the diagonals, d₁d₂48 cm²
- Area of one of the four right triangles, d₁d₂/86 cm²
- Ratio of the diagonals1.333
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.
- Diagonals of 8 cm and 6 cm give an area of 24 cm², half of the 48 cm² rectangle that surrounds them.
- Doubling either diagonal doubles the area, while doubling both quadruples it.
Chart data as tables
Ranked variables
| Variable | Typical range | Lowest result | Highest result | Swing (cm²) | Local step | Change in result |
|---|---|---|---|---|---|---|
| d2 Second diagonal | 1 to 40 cm | 4 | 160 | 156 | +10% | +10 % |
| d1 First diagonal | 1 to 40 cm | 3 | 120 | 117 | +10% | +10 % |
Sweep of First diagonal
| First diagonal (cm) | Area (cm²) |
|---|---|
| 1 | 3 |
| 5.875 | 17.63 |
| 10.75 | 32.25 |
| 15.63 | 46.88 |
| 20.5 | 61.5 |
| 25.38 | 76.13 |
| 30.25 | 90.75 |
| 35.13 | 105.4 |
| 40 | 120 |
Sweep one variable
Variables and how each one contributes
Area
What the equation gives you. Everything else feeds it.
First diagonal
The full length of the first diagonal, in cm. A longer diagonal gives a larger area in direct proportion. Raising it by 10 % (to 8.8) moves the result by +10 %, from 24 to 26.4 cm².
Second diagonal
The full length of the second diagonal, in cm. It crosses d₁ at a right angle. A longer diagonal gives a larger area in direct proportion. Raising it by 10 % (to 6.6) moves the result by +10 %, from 24 to 26.4 cm².
Worked example with the values above
- The diagonals are d₁ = 8 cm and d₂ = 6 cm.
- Multiply them: 8 × 6 = 48.
- Halve it: 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