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The radius of this Apollonius circle is + where is the incircle radius and is the semiperimeter of the triangle. [ 27 ] The following relations hold among the inradius r {\displaystyle r} , the circumradius R {\displaystyle R} , the semiperimeter s {\displaystyle s} , and the excircle radii r a {\displaystyle r_{a}} , r b {\displaystyle r_{b ...
Using radians, the formula for the arc length s of a circular arc of radius r and subtending a central angle of measure 𝜃 is =, and the formula for the area A of a circular sector of radius r and with central angle of measure 𝜃 is A = 1 2 θ r 2 . {\displaystyle A={\frac {1}{2}}\theta r^{2}.}
Note that as a matter of convention, the previous calculations excluded the circle of radius corresponding to the fraction . It includes the complete circle for 1 1 {\displaystyle {\frac {1}{1}}} , half of which lies outside the unit interval, hence the sum is still the fraction of the unit square covered by Ford circles.
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.
Perimeter#Formulas – Path that surrounds an area; List of second moments of area; List of surface-area-to-volume ratios – Surface area per unit volume; List of surface area formulas – Measure of a two-dimensional surface; List of trigonometric identities; List of volume formulas – Quantity of three-dimensional space
as one would expect. This is equivalent to the above definition of the 2D mean diameter. However, for historical reasons, the hydraulic radius is defined as the cross-sectional area of a pipe A, divided by its wetted perimeter P, which leads to =, and the hydraulic radius is half of the 2D mean radius. [3]