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The total area of a circle is πr 2. The area of the sector can be obtained by multiplying the circle's area by the ratio of the angle θ (expressed in radians) and 2π (because the area of the sector is directly proportional to its angle, and 2π is the angle for the whole circle, in radians): = =
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 ): a = R 2 2 ( θ − sin θ ) {\displaystyle a={\tfrac {R^{2}}{2}}\left(\theta -\sin \theta \right)}
Shape Figure ¯ ¯ Area rectangle area: General triangular area + + [1] Isosceles-triangular area: Right-triangular area: Circular area: Quarter-circular area [2]: Semicircular area [3]: Circular sector
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}.}
Since the area of a circular sector with radius r and angle u (in radians) is r 2 u/2, it will be equal to u when r = √ 2. In the diagram, such a circle is tangent to the hyperbola xy = 1 at (1,1). The yellow sector depicts an area and angle magnitude.
One radian corresponds to the angle for which s = r, hence 1 radian = 1 m/m = 1. [9] However, rad is only to be used to express angles, not to express ratios of lengths in general. [7] A similar calculation using the area of a circular sector θ = 2A/r 2 gives 1 radian as 1 m 2 /m 2 = 1. [10] The key fact is that the radian is a dimensionless ...
One radian corresponds to the angle for which s = r, hence 1 radian = 1 m/m = 1. [33] However, rad is only to be used to express angles, not to express ratios of lengths in general. [34] A similar calculation using the area of a circular sector θ = 2A/r 2 gives 1 radian as 1 m 2 /m 2 = 1. [35] The key fact is that the radian is a dimensionless ...
If P 0 is taken to be the point (1, 1), P 1 the point (x 1, 1/x 1), and P 2 the point (x 2, 1/x 2), then the parallel condition requires that Q be the point (x 1 x 2, 1/x 1 1/x 2). It thus makes sense to define the hyperbolic angle from P 0 to an arbitrary point on the curve as a logarithmic function of the point's value of x .