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An elongated shape can be made more round while keeping its perimeter fixed and increasing its area. The classical isoperimetric problem dates back to antiquity. [2] The problem can be stated as follows: Among all closed curves in the plane of fixed perimeter, which curve (if any) maximizes the area of its enclosed region? This question can be ...
The isoperimetric problem is to determine a figure with the largest area, amongst those having a given perimeter. The solution is intuitive; it is the circle. In particular, this can be used to explain why drops of fat on a broth surface are circular. This problem may seem simple, but its mathematical proof requires some sophisticated theorems.
For shapes with curved boundary, calculus is usually required to compute the area. Indeed, the problem of determining the area of plane figures was a major motivation for the historical development of calculus. [5] For a solid shape such as a sphere, cone, or cylinder, the area of its boundary surface is called the surface area.
The napkin folding problem is a problem in geometry and the mathematics of paper folding that explores whether folding a square or a rectangular napkin can increase its perimeter. The problem is known under several names, including the Margulis napkin problem , suggesting it is due to Grigory Margulis , and the Arnold's rouble problem referring ...
Alternatively, the shape's area could be compared to that of its bounding circle, [1] [2] its convex hull, [1] [3] or its minimum bounding box. [3] Similarly, a comparison can be made between the perimeter of the shape and that of its convex hull, [3] its bounding circle, [1] or a circle having the same area. [1]
Shoelace scheme for determining the area of a polygon with point coordinates (,),..., (,). The shoelace formula, also known as Gauss's area formula and the surveyor's formula, [1] is a mathematical algorithm to determine the area of a simple polygon whose vertices are described by their Cartesian coordinates in the plane. [2]
Let p be an interior point of the disk, and let n be a multiple of 4 that is greater than or equal to 8. Form n sectors of the disk with equal angles by choosing an arbitrary line through p, rotating the line n / 2 − 1 times by an angle of 2 π / n radians, and slicing the disk on each of the resulting n / 2 lines.
The area of a regular polygon is half its perimeter multiplied by the distance from its center to its sides, and because the sequence tends to a circle, the corresponding formula–that the area is half the circumference times the radius–namely, A = 1 / 2 × 2πr × r, holds for a circle.