KEVOS® MAKEOS · Geometry
Area of an Irregular Outline by Simpson’s Rule
What is the area of this irregular outline from measured heights at equal spacing?
Area of a plane figure of irregular outline by Simpson’s rule from up to eleven equally spaced ordinates (heights), with the trapezoidal-rule result for comparison; also gives the volume when the ordinates are cross-section areas.
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How this tool works
Inputs, units and where to find them
| Input | Unit | Typical source |
|---|---|---|
| Strip width (spacing) | mm | Equal width of each strip. |
| Number of strips (even) | Number of strips; Simpson’s rule needs an even number. | |
| Heights V₀ … V₁₀ (only V₀ to V_N are used) | ||
| V0 | mm | Height of the figure at strip boundary 0 (or the section area for a volume). |
| V1 | mm | Height of the figure at strip boundary 1 (or the section area for a volume). |
| V2 | mm | Height of the figure at strip boundary 2 (or the section area for a volume). |
| V3 | mm | Height of the figure at strip boundary 3 (or the section area for a volume). |
| V4 | mm | Height of the figure at strip boundary 4 (or the section area for a volume). |
| V5 | mm | Height of the figure at strip boundary 5 (or the section area for a volume). |
| V6 | mm | Height of the figure at strip boundary 6 (or the section area for a volume). |
| V7 | mm | Height of the figure at strip boundary 7 (or the section area for a volume). |
| V8 | mm | Height of the figure at strip boundary 8 (or the section area for a volume). |
| V9 | mm | Height of the figure at strip boundary 9 (or the section area for a volume). |
| V10 | mm | Height of the figure at strip boundary 10 (or the section area for a volume). |
Method and formulas
- A = (W ÷ 3) [(V₀ + V_N) + 4 (V₁ + V₃ + … + V_N−1) + 2 (V₂ + V₄ + … + V_N−2)].
- Trapezoidal check: A = W [(V₀ + V_N) ÷ 2 + V₁ + … + V_N−1].
- Entering section areas in place of heights gives a volume.
- Example: eight strips 10 mm wide give an area of 1,266.7 mm².
Limitations
- Exact for curves up to the third degree between ordinates; accuracy improves with more strips.
- Strips must be of equal width.
- Example values are illustrations. Replace them with values for your actual task.
Typical uses
- Find the area of a cam, template or casting outline from a drawing
- Find a volume from equally spaced cross-section areas
- Integrate a measured curve such as a pressure or force trace
Who it is for
- Engineers
- Technicians
- Engineering students
What you get
- Area by Simpson’s rule
- Area by the trapezoidal rule
- Difference between the two
Interpreting the result
- A large difference from the trapezoidal result means more strips are needed.
References
- Machinery’s Handbook, 29th edition (Industrial Press, 2012), printed pp. 67–68. Topic reference; no source tables or recommended settings are reproduced.
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