Subtended angle: Difference between revisions

Latest revision as of 11:52, 15 February 2025

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

Subtended Angle[edit]

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.

Definition[edit]

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[edit]

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[edit]

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[edit]

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[edit]

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-'''Subtended angle''' refers to the angle formed at a specific point when two lines or line segments extend from that point and meet at another point on a circle or any other curve. This concept is fundamental in the fields of [[geometry]], [[trigonometry]], and various applications in [[physics]], [[engineering]], and [[astronomy]]. Understanding subtended angles is crucial for solving problems related to circles and arcs, as well as for applications in real-world scenarios such as determining the position of celestial bodies or designing mechanical systems.
+{{Short description|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.
+[[File:Subtended_angle.svg|thumb|right|Diagram illustrating a subtended angle.]]
 ==Definition==
-A subtended angle is defined as the angle formed at a particular point when two lines originating from that point intersect a curve or a circle. In the context of a circle, if you have a circle with a center ''O'', and two points ''A'' and ''B'' on the circumference of the circle, the angle ∠AOB is said to be subtended by the arc ''AB'' at the center. Similarly, if you observe the angle from a point on the circumference, such as point ''A'' looking towards another point ''B'' across the circle, the angle subtended by the arc ''AB'' at point ''A'' is different from the angle subtended at the center ''O''.
+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==
-One of the key properties of subtended angles is that any angle subtended by an arc at the center of the circle is twice the size of the angle subtended by the same arc at any point on the circumference of the circle. This property is fundamental in the study of circle theorems and has various applications in solving geometrical problems.
+* 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.
-Another important property is that angles subtended by the same arc at the circumference of the circle are equal. This is known as the ''Angle in the Same Segment Theorem'' and is often used to prove that certain lines are parallel or to calculate unknown angles in geometric figures.
+* 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 have numerous applications across different fields. In [[astronomy]], they are used to calculate the apparent sizes of celestial bodies and their distances from Earth. In [[engineering]] and [[architecture]], understanding subtended angles is crucial for designing curved structures and components, such as bridges, arches, and gears. In [[physics]], subtended angles play a role in the analysis of phenomena such as diffraction and the calculation of angular velocity.
+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.
-==Calculating Subtended Angles==
+==Related Concepts==
-The calculation of subtended angles involves various formulas, depending on the context and the known parameters. For a circle, if the length of the arc ''AB'' and the radius ''r'' of the circle are known, the subtended angle ∠AOB at the center can be calculated using the formula:
+* [[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.
-\[ \text{Angle} = \frac{\text{Arc Length}}{\text{Radius}} \times \frac{180}{\pi} \]
+==Related pages==
+* [[Angle]]
-where the angle is measured in degrees.
-==See Also==
 * [[Circle]]
-* [[Arc (geometry)|Arc]]
+* [[Arc (geometry)]]
-* [[Central angle]]
+* [[Geometry]]
-* [[Inscribed angle]]
-* [[Angle in the Same Segment Theorem]]
-==References==
-<references/>
 [[Category:Geometry]]
-[[Category:Mathematical concepts]]
-{{geometry-stub}}