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In geometry, the circumference (from Latin circumferens, meaning "carrying around") is the perimeter of a circle or ellipse. The circumference is the arc length of the circle, as if it were opened up and straightened out to a line segment. [1] More generally, the perimeter is the curve length around any closed figure.
The circle is the shape with the largest area for a given length of perimeter (see Isoperimetric inequality). The circle is a highly symmetric shape: every line through the centre forms a line of reflection symmetry , and it has rotational symmetry around the centre for every angle.
A perimeter is a closed path that encompasses, surrounds, or outlines either a two dimensional shape or a one-dimensional length. The perimeter of a circle or an ellipse is called its circumference. Calculating the perimeter has several practical applications. A calculated perimeter is the length of fence required to surround a yard or garden.
Circle with similar triangles: circumscribed side, inscribed side and complement, inscribed split side and complement. Let one side of an inscribed regular n-gon have length s n and touch the circle at points A and B. Let A′ be the point opposite A on the circle, so that A′A is a diameter, and A′AB is an inscribed triangle on a diameter.
Proposition one states: The area of any circle is equal to a right-angled triangle in which one of the sides about the right angle is equal to the radius, and the other to the circumference of the circle. Any circle with a circumference c and a radius r is equal in area with a right triangle with the two legs being c and r.
The area within a circle is equal to the radius multiplied by half the circumference, or A = r x C /2 = r x r x π.. Liu Hui argued: "Multiply one side of a hexagon by the radius (of its circumcircle), then multiply this by three, to yield the area of a dodecagon; if we cut a hexagon into a dodecagon, multiply its side by its radius, then again multiply by six, we get the area of a 24-gon; the ...
The arc length, from the familiar geometry of a circle, is s = θ R {\displaystyle s={\theta }R} The area a of the circular segment is equal to the area of the circular sector minus the area of the triangular portion (using the double angle formula to get an equation in terms of θ {\displaystyle \theta } ):
where C is the circumference of a circle, d is the diameter, and r is the radius. More generally, = where L and w are, respectively, the perimeter and the width of any curve of constant width. = where A is the area of a circle. More generally, =