The Mysterious Perimeter of the Circle

The formula for the perimeter of a circle is 2 π r where “r” is the radius. Since there are 360 degrees in a full circle, it follows that 360 ÷ 2π gives an approximation for the area of a circle. The formula for perimeter of the circle can also be used to find a length around any shape that is circular in plan, such as a square with rounded corners.

In mathematics, Archimedes’ constant or pi-squared, usually denoted by the Greek letter π (lower case pi), is the irrational mathematical constant ‏π‏ that represents the ratio of a circle’s circumference to itsdiameter.

Pi-squared is also equal to half of 100.
Archimedes’ constant was first calculated in the 16th century when, in an elaborate investigation involving polygons inscribed within circles, François Viète approximated this value by inscribing and then circumscribing a 96-sided polygon around a circle; this yielded 250/153 = 1535/780 = 3.141855…, which rounds to 3.1416 (the term “3.” denotes “rounded down”). Later in the 17th century, William Neile arrived at a more accurate value of 3.1415929 by inscribing several polygons within the circle, the method also devised by Archimedes.
The fractional part of ‏π‏ is called the tangent.
The term perimeter may be derived from the perimeter, which means “to measure round something,” or it may come from the Hellenistic Greek περιμέτρος, meaning circumference. The perimeter of a circle is the length around it.
Perimeter is usually called the “length” of a closed curve or figure, but sometimes also refers to the length measured along any closed path in a plane that includes both endpoints.
While individual instances of lengths are always positive (one cannot walk on negative distance), paths can be extended to form complex shapes that may approach something like an “infinite perimeter” where, for example, one traverses parts of some regular polygon outside its interior points.
In infinitary logic, an infinite perimeter implies infinite area. However, this does not hold in the real world; for example, a circle is defined as having a circumference of 2πr. Because this formula contains an infinite summation, applying it to measure the perimeter of a finite-sized circle will yield an expression that involves π to an infinite power, yielding an incorrect result.
In geometry, the perimeter is often used in problems related to circles and spheres. For instance, in solving for the area enclosed by a circular ring or spherical shell with known diameter “D,” one computes directly ‏(2π D) without using any formulas for circumference. This can be seen by cutting off two small pieces from the top left and bottom right sections of such region, which reduces them to triangles whose areas are base × height.
For more general shapes, the perimeter is equal to the sum of the lengths of all of its line segments and curves that form the boundary (the length that would be measured with a “tape”). This applies regardless of whether these line segments and curves are straight or curved.
The total length is given by “perimeter” = 2 * (“length” + “width”), which may be thought of as measuring around it in two possible directions, e.g., if “length” was defined as breadth, then perimeter would be twice width instead.
Perimeters are studied in both elementary geometries, where they are seen as defining areas with flat shapes. Perimeters are part of integral calculus, giving the length of line segments, curves s and surfaces.
When calculating the perimeter of a polygon (one closed shape), there is always at least one right angle (a 90° angle ). If all angles are less than 90°, then it is called an orthodiagonal polygon. The longest side of the triangle intersects that diagonal, dividing it into two triangles, thus determining half of the perimeter.