Subtended angle

Example of the angles subtended by an arc from two points

In geometry, a side of a triangle subtends (from Latin for "stretch under") the opposite angle, and in particular a chord of a circle subtends the central angle formed by the radii through its endpoints. The same concept has been generalized beyond line segments, and an arc of a curve is said to subtend an angle if its endpoints lie on the two rays forming the angle. A chord of a circle or arbitrary curve is also said to subtend the corresponding arc of the curve.

If a straight or curved segment subtends an angle, the angle is subtended by the segment. Sometimes the term subtend is applied in the opposite sense, and the angle is said to subtend the segment. Alternately the angle can be said to intercept or enclose the segment.

Generalized to three-dimensional space, a surface subtends a solid angle if its boundary defines the cone of the angle.

Many theorems in geometry relate to subtended angles. If two sides of a triangle are congruent, then the angles they subtend are also congruent, and conversely if two angles are congruent then they are subtended by congruent sides (propositions I.5–6 in Euclid's Elements), forming an isosceles triangle. More generally, the law of sines states that the sine of each angle of a triangle is proportional to the side subtending it. The inscribed angle theorem states that when the vertex of an angle inscribed in a circle lies on the same side of the chord subtending it as the center of the circle, then the central angle subtended by the same chord is twice the inscribed angle.

External links

• Definition of subtended angle, mathisfun.com, with interactive applet
• How an object subtends an angle, Math Open Reference, with interactive applet
• Angle definition pages, Math Open Reference, with interactive applets that are also useful in a classroom setting.