Subtended angle

An angle formed by two lines originating from a common point and intersecting a curve or surface.

Subtended Angle

A subtended angle is an angle formed at a specific point by two lines or rays that originate from that point and intersect a curve or surface. This concept is commonly used in geometry and trigonometry to describe the angle formed by an arc, a segment, or a chord of a circle at a particular point on the circle or outside it.

Diagram illustrating a subtended angle.

Definition

In the context of a circle, a subtended angle is the angle formed at the center of the circle by two radii that extend to the endpoints of an arc. Alternatively, it can also refer to the angle formed at any point on the circle by two lines that intersect the circle at the endpoints of a chord.

Properties

• The subtended angle at the center of a circle is twice the angle subtended at any point on the circumference by the same arc. This is known as the inscribed angle theorem.
• If two angles subtend the same arc or chord, they are equal.
• The angle subtended by a diameter of a circle at any point on the circle is a right angle, according to Thales' theorem.

Applications

Subtended angles are used in various fields such as astronomy, optics, and navigation. In astronomy, the concept helps in determining the apparent size of celestial objects. In optics, it is used to calculate the field of view of lenses and mirrors.

Related Concepts

• Central angle: An angle whose vertex is the center of a circle and whose sides are radii.
• Inscribed angle: An angle formed by two chords in a circle which have a common endpoint.
• Chord (geometry): A straight line segment whose endpoints both lie on the circle.

Related pages

• Angle
• Circle
• Arc (geometry)
• Geometry