Isoperimetric Area Budget
Spend an equal perimeter on different shapes and compare the enclosed areas.
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Isoperimetric Area Budget
The isoperimetric theorem states that among all planar shapes with a given perimeter, the circle encloses the maximum possible area.
Worked example: Perimeter 2
Given a perimeter P = 2, we can examine the area of different regular polygons. The area An of a regular n-gon is given by An = P2 / (4n tan(π/n)).
- For n = 4 (a square), the area is 22 / (16 tan(π/4)) = 4 / 16 = 0.25.
- The circle bound for P = 2 is P2 / (4π) = 4 / (4π) = 1/π ≈ 0.3183.
- The isoperimetric quotient is the ratio of the shape's area to the circle's area, which is 0.25 / (1/π) = π / 4 ≈ 0.7854.
| Shape | Area | Circle bound |
|---|---|---|
| Square (n=4) | 0.2500 | 0.3183 |
Compare areas for a fixed perimeter
Open Isoperimetric Area Budget to experiment with different shapes.
- Set the Perimeter to 2 and Shape to 4 (a square). Observe the area is 0.25 and the area deficit from a circle is about 0.0683.
- Change the Shape to a Circle. Notice the area reaches the circle bound of ≈ 0.3183, and the deficit drops to 0. The quotient is exactly 1.
- Increase the Perimeter to 4 and choose Shape 6 (hexagon). The area is ≈ 1.1547 compared to a circle bound of ≈ 1.2732.
Limitations and general curves
The continuous plane isoperimetric theorem applies to simple closed rectifiable curves. While the regular polygons demonstrate that area increases as the number of sides grows, they do not constitute a formal proof of the theorem for all general curves.
Additionally, a degenerate "collapsed segment" (where a shape travels out and back along a line segment) uses its entire perimeter budget P without enclosing any area. In the module, this limiting probe yields an area of 0, an isoperimetric quotient of 0, and a maximal deficit. It is not a simple Jordan curve.
References
For further reading, please refer to the following sources which detail the mathematical principles and proofs discussed in this module. The sources provide a comprehensive background on the continuous isoperimetric inequality, its geometric principles, historical developments in the field of calculus of variations, and detailed investigations into its discrete analogues for polygonal domains.
- arXiv:1909.06347: Background on the continuous isoperimetric inequality and its discrete analogues.
Experience it interactively
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