Perimeter
Perimeter is the total length (or the arc length) along the border or outer boundary of a closed two-dimensional plane or curve. For example, the perimeter of a circle is the total length around its boundary, while the perimeter of a polygon is the total length (or sum) of its sides. The word perimeter is derived from the Greek words "peri" (around) and "metron" (to measure).
The perimeter of a circle is commonly called the circle's circumference. With regards to the radius, r, or diameter, d, of a circle, the equation to solve for perimeter, p, is p = 2r = d, where pi, , is a constant number that is defined as the ratio of a circle's perimeter (or circumference) to its diameter, with its value approximately equal to 3.14159. As an example, if the radius of a circle is measured to be 20 centimeters and the approximate value of 3.14 is used for pi, then the perimeter is calculated to be "2 x 3.14 x 20 cm", or p = 125.6 cm.
Polygons are defined as closed planar figures with straight sides. Vertices (or edges) are the means by which polygons are classified. For instance, rectangles (with four vertices) and triangles (with three vertices) are particular types of polygons. The perimeter of any polygon is the total length of its sides.
The general equation for determining the perimeter of a rectangle is "p = 2W + 2L", where "W" is the rectangle's width and "L" is the rectangle's length. Knowing that a rectangle has four sides, with opposite sides of the rectangle having the same length, the perimeter of a rectangle having side-lengths of 3.4 cm and 8.2 cm can be solved by realizing that the rectangle has two sides of length 3.4 cm, and two remaining sides of length of 8.2 cm. Therefore, the sum of the lengths of all four sides of the rectangle is "3.4 + 3.4 + 8.2 + 8.2 = 23.2 cm". The perimeter of the rectangle is p = 23.2 cm.
Triangles are classified in terms of their sides and angles. Equilateral triangles have three equal sides, isosceles triangles have two equal sides, and scalene triangles have no equal sides. In acute triangles, angles are less than ninety degrees; in right triangles, one angle is equal to ninety degrees; and in obtuse triangles, one angle is greater than ninety degrees. But, with regards to all triangles, the perimeter of a triangle is calculated exactly the same way: by measuring its three sides. For example, if a triangle has side "a" with length 3 cm, side "b" with length 4 cm, and side "c" with length 5 cm, then the length of the triangle's perimeter, p, is "3 + 4 + 5". Therefore p = 12 cm.
Determining the perimeter of a bounded curve can be accomplished by referring to geometry that the circumference of a circle is defined as the limit of the perimeters of regular polygons inscribed in the circle. A similar method is used for a bounded curve. For a portion of the curve from the point (x1, Φ(x1)) to the point (x2, Φ(x2)) the formula for the distance between these two points is given by . This distance formula between those two points can then be expanded over the entire curve to solve (with the use of calculus) the perimeter of the bounded curve. The tools of calculus necessary to determine this integration are deferred to the calculus-related articles.
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