Angle Subtended by Arc at Center
The angle subtended by an arc at the centre of a circle is the angular measure created by two radii commencing from the centre and continuing to the arc’s ends. It is the basic connection between the central angle and the appropriate arc length. This angle, given in radians or degrees, controls the length of the arc, with a direct ratio to the radius.
Theorem of Angle Subtended by Arc at Center
Given: An arc PQ of a circle is given, bounded by angles ∠POQ at the centre O and ∠PAQ at a point A on the remaining half of the circle.
To prove : ∠POQ=2∠PAQ
Construction : Join the line that AO extended to B.
Proof:
∠BOQ=∠OAQ+∠AQO . . . (1)
Also, in △ OAQ,
OA=OQ [The radius of a circle]
Therefore,
∠OAQ=∠OQA [Angles opposing equal sides have the same value]
∠BOQ=2∠OAQ . . . (2)
Similarly, BOP=2∠OAP . . . (3)
Adding 2 & 3, we get,
∠BOP+∠BOQ=2(∠OAP+∠OAQ)
∠POQ=2∠PAQ . . . (4)
For instance 3, where PQ is the main arc, equation 4 is substituted by Reflex angle, ∠POQ=2∠PAQ.
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Arc of a Circle
Arc of a Circle is a part of the circumference of a circle or we can also say the Arc of a Circle is some percentage of the circle’s circumference. As we know, a circle is defined as a two-dimensional geometrical object where all the points are equidistant from the center and the distance measured around the circle is known as a circumference and some portion of the circumference taken at a time is known as the Arc of a Circle.
In this article, we will learn the Arc of a Circle in detail, including its definition, types, and arc length formula. Other than that we will also discuss the angle subtended by an arc and the theorem related to this angle as well.
Table of Content
- What is the Arc of a Circle?
- Types of Arcs
- Arc of the Circle Formula
- How to Find Length of Arc of a Circle?
- Angle Subtended by Arc at Center