BIZOMAT 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). |
What each input means
Open an input to see what it is, its units, the worked example and the common mistakes, with a small diagram.
What this means: Strip width (spacing)
- What it is
- The equal spacing between neighbouring ordinates along the base line of the outline: the overall length divided by the number of strips.
- Units and format
- Enter a number in mm: millimetres. The form accepts 0.000001 mm to 10000000 mm. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 10 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Effect on the result
- Both the Simpson and trapezoidal areas are directly proportional to the strip width.
- Get it right
- All strips must be the same width; Simpson’s rule does not work with uneven spacing.
What this means: Number of strips (even)
- What it is
- How many equal strips the base line is divided into. It sets how many ordinates are read: one more than the number of strips, from V0 to V at that number.
- Units and format
- Enter a plain number with no unit. The form accepts 2 to 10. Use a whole number. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 8 as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Simpson’s rule needs an even number; an odd count is rejected. Ordinates beyond the last one used are ignored.
Heights V₀ … V₁₀ (only V₀ to V_N are used)
What this means: V0
- What it is
- The ordinate at the start of the base line (boundary 0): the height of the outline measured square to the base line, or the section area there when finding a volume.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- A shape that starts at a point has a first height of zero; the first and last ordinates each count once.
What this means: V1
- What it is
- The ordinate at boundary 1, one strip width from the start: the height of the outline square to the base line, or the section area at that station for a volume.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Odd-numbered ordinates carry a weight of four, so measure them carefully.
What this means: V2
- What it is
- The ordinate at boundary 2, two strip widths from the start, measured square to the base line (or a section area when finding a volume).
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 18 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Even-numbered interior ordinates carry a weight of two; if two strips are used this is the last ordinate and counts once.
What this means: V3
- What it is
- The ordinate at boundary 3, three strip widths from the start, measured square to the base line (or a section area when finding a volume).
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 21 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Odd-numbered ordinates carry a weight of four. Below the base line a height is negative.
What this means: V4
- What it is
- The ordinate at boundary 4, four strip widths from the start, measured square to the base line (or a section area when finding a volume).
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 22 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when fewer than four strips are used; it is the last ordinate when there are exactly four.
What this means: V5
- What it is
- The ordinate at boundary 5, five strip widths from the start, measured square to the base line (or a section area when finding a volume).
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 21 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only with six or more strips, where it carries a weight of four.
What this means: V6
- What it is
- The ordinate at boundary 6, six strip widths from the start, measured square to the base line (or a section area when finding a volume).
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 18 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when fewer than six strips are used; it is the last ordinate when there are exactly six.
What this means: V7
- What it is
- The ordinate at boundary 7, seven strip widths from the start, measured square to the base line (or a section area when finding a volume).
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 12 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only with eight or more strips, where it carries a weight of four.
What this means: V8
- What it is
- The ordinate at boundary 8, eight strip widths from the start, measured square to the base line (or a section area when finding a volume).
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Ignored when fewer than eight strips are used; it is the last ordinate when there are exactly eight.
What this means: V9
- What it is
- The ordinate at boundary 9, nine strip widths from the start, used only when the base is divided into ten strips.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Leave it at zero unless ten strips are used; otherwise it is ignored.
What this means: V10
- What it is
- The ordinate at boundary 10, the end of the base line when ten strips are used: the height of the outline there, or the section area for a volume.
- Units and format
- Enter a number in mm: millimetres. Negative values are accepted. Required: the tool does not calculate until every input is valid.
- Example
- The form opens with 0 mm as a worked example. It is an illustration, not a recommended value: replace it with your own figure.
- Get it right
- Used only with ten strips. A shape that ends at a point has a last height of zero.
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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