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A circle circumference and radius are proportional. The area enclosed and the square of its radius are proportional. The constants of proportionality are 2 π and π respectively. The circle that is centred at the origin with radius 1 is called the unit circle. Thought of as a great circle of the unit sphere, it becomes the Riemannian circle.
The radius of the incircle is related to the area of the triangle. [18] The ratio of the area of the incircle to the area of the triangle is less than or equal to π / 3 3 {\displaystyle \pi {\big /}3{\sqrt {3}}} , with equality holding only for equilateral triangles .
In the following equations, denotes the sagitta (the depth or height of the arc), equals the radius of the circle, and the length of the chord spanning the base of the arc. As 1 2 l {\displaystyle {\tfrac {1}{2}}l} and r − s {\displaystyle r-s} are two sides of a right triangle with r {\displaystyle r} as the hypotenuse , the Pythagorean ...
The hydraulic diameter is the equivalent circular configuration with the same circumference as the wetted perimeter. The area of a circle of radius R is . Given the area of a non-circular object A, one can calculate its area-equivalent radius by setting = or, alternatively:
Radius of curvature and center of curvature. In differential geometry, the radius of curvature, R, is the reciprocal of the curvature. For a curve, it equals the radius of the circular arc which best approximates the curve at that point. For surfaces, the radius of curvature is the radius of a circle that best fits a normal section or ...
To calculate a circle's perimeter, knowledge of its radius or diameter and the number π suffices. The problem is that π is not rational (it cannot be expressed as the quotient of two integers ), nor is it algebraic (it is not a root of a polynomial equation with rational coefficients).