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A circular sector is shaded in green. Its curved boundary of length L is a circular arc. A circular arc is the arc of a circle between a pair of distinct points.If the two points are not directly opposite each other, one of these arcs, the minor arc, subtends an angle at the center of the circle that is less than π radians (180 degrees); and the other arc, the major arc, subtends an angle ...
Using radians, the formula for the arc length s of a circular arc of radius r and subtending a central angle of measure π is =, and the formula for the area A of a circular sector of radius r and with central angle of measure π is A = 1 2 θ r 2 . {\displaystyle A={\frac {1}{2}}\theta r^{2}.}
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 degree of curvature is based on 100 units of arc length, the conversion between degree of curvature and radius is Dr = 18000/π ≈ 5729.57795, where D is degree and r is radius. Since rail routes have very large radii, they are laid out in chords, as the difference to the arc is inconsequential; this made work easier before electronic ...
In the following lines, represents the radius of a circle, is its diameter, is its circumference, is the length of an arc of the circle, and is the angle which the arc subtends at the centre of the circle.
Another proof that uses triangles considers the area enclosed by a circle to be made up of an infinite number of triangles (i.e. the triangles each have an angle of dπ at the centre of the circle), each with an area of β 1 / 2 β · r 2 · dπ (derived from the expression for the area of a triangle: β 1 / 2 β · a · b · sinπ ...
Since a radian is mathematically defined as the angle formed when the length of a circular arc equals the radius of the circle, a milliradian, is the angle formed when the length of a circular arc equals β 1 / 1000 β of the radius of the circle.
In geometry, the sagitta (sometimes abbreviated as sag [1]) of a circular arc is the distance from the midpoint of the arc to the midpoint of its chord. [2] It is used extensively in architecture when calculating the arc necessary to span a certain height and distance and also in optics where it is used to find the depth of a spherical mirror ...