Full-Option Calculator
Example data table
Typical rectangle input and common matching outputs.
| Shape | Inputs | Equal area side | Equal perimeter side | Equal Dh side |
|---|---|---|---|---|
| Rectangle | W=200 mm, H=100 mm | 141.421 mm | 150 mm | 133.333 mm |
Formula used
This calculator derives an equivalent square side s from a chosen matching rule.
Ellipse perimeter uses a Ramanujan approximation for better accuracy without iterations.
How to use this calculator
- Select the original shape and enter its dimensions.
- Choose a matching rule (area, perimeter, hydraulic, or inertia).
- Pick your unit system for consistent inputs and outputs.
- Press Compute Equivalent Square to see results above.
- Use Download CSV or Download PDF for reporting.
A concise technical overview of equivalent-square mapping in physics and engineering.
1) Why Equivalent Squares Matter in Physics
Many transport and structural models accept square cross-sections as a baseline. An equivalent square provides one representative length, letting you compare different geometries while keeping a chosen physical property unchanged.
2) Choosing the Right Matching Rule
There is no single “best” equivalence; the rule should match your dominant physics. Use area when mass storage or volumetric throughput matters, perimeter when wall interaction dominates, hydraulic diameter for internal-flow similarity, and inertia when bending stiffness is the priority. In practice, engineers often compute two rules to bracket expected behavior.
3) Area Matching: Conserving Storage and Mass Flux
Area matching uses s = √A, so the square has the same cross-sectional area as the original. For a 200 mm × 100 mm rectangle, the equivalent-area side is 141.421 mm, preserving flux definitions like flow rate per area.
4) Perimeter Matching: Surface Interaction and Boundary Effects
Perimeter matching sets s = P/4, keeping boundary length constant. It helps when wall-driven effects scale with wetted perimeter. For a 200 mm × 100 mm rectangle, the equal-perimeter side is 150 mm.
5) Hydraulic Diameter Matching: Flow and Convection Scaling
Internal flow often uses Dh = 4A/P. For a square, Dh = s, so the equivalent side becomes s = 4A/P. With 200 mm × 100 mm, the equal-Dh side is 133.333 mm, frequently closer to convection correlations than area matching.
6) Inertia Matching: Stiffness and Bending Comparisons
When comparing bending behavior, matching the second moment of area can be more meaningful than matching size alone. For a square about its centroid, I = s4/12, giving s = (12I)1/4. This is useful for approximate stiffness equivalence across different shapes and axes.
7) Handling Circles and Ellipses Reliably
Circles have clean expressions for area, perimeter, and inertia, enabling fast equivalence checks for pipes and rods. Ellipses require a perimeter estimate; this calculator uses a Ramanujan-style approximation to avoid iterative solvers while remaining well behaved across common aspect ratios used in ducts and apertures.
8) Reporting, Traceability, and Unit Consistency
Different rules produce different sides, so reports should state the matching rule with the computed s. Keep inputs in one unit system; exports record the shape, rule, and derived metrics for lab notes and reviews.
FAQs
1) What is an equivalent square?
An equivalent square is a square cross-section whose side length is chosen so one selected property (area, perimeter, hydraulic diameter, or inertia) matches the original shape for fair comparison.
2) Which rule should I use for duct flow?
Hydraulic diameter matching is typically the best first choice for internal flow and convection correlations, because many dimensionless groups and empirical fits are written in terms of Dh.
3) Why do equal area and equal hydraulic diameter give different sides?
Area depends on the cross-section size, while hydraulic diameter depends on both area and perimeter. Two shapes with the same area can have different wall contact, changing 4A/P and therefore the equivalent side.
4) Can I use this for irregular sections?
Yes. Choose “Custom” and enter the measured area and perimeter (and inertia if needed). This supports lab-derived sections or CAD measurements when a closed-form formula is not available.
5) What units should I enter?
Select one length unit and enter all dimensions consistently in that unit. The calculator converts internally and returns results in the same unit, including squared and fourth-power quantities where applicable.
6) How is ellipse perimeter handled?
Ellipse perimeter has no simple exact elementary formula. The calculator uses a widely used Ramanujan-type approximation, which provides stable, accurate results for typical engineering aspect ratios without slow iterations.
7) Do the CSV and PDF exports include my inputs?
Yes. Exports include the shape description, matching rule, unit, original metrics, and the equivalent square outputs. This makes it easy to paste results into reports or attach them to lab records.