Subtended angle

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Illustration of the angles subtended by an arc from two points

In geometry, an angle subtended by an arc, line, or other curve is one whose two rays pass through the endpoints of the arc. The precise meaning varies with the context. For example, one may speak of the angle subtended by an arc of a circumference when the angle's vertex is a point on the circumference. A simple theorem of plane geometry states that arcs of equal lengths subtend equal angles in such a situation.

Similarly, a surface or solid object subtends a solid angle, which is the area on a unit sphere cut out by the envelope of the vectors defining the perimeter of the surface or object. The solid angle subtended by a circular cone of opening angle (aperture) θ is

2 \pi \left (1 - \cos {\theta \over 2} \right)\,\mathrm{sr}.

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