Angle Subtended by an Arc of a Circle: Definitions, Theorems, Solved Examples

Angle Subtended by an Arc of a Circle: A circle is a closed figure round in shape. The part of the boundary or circumference of a circle is called an arc of the circle. There are two types of arcs formed in the circle, viz. major arc and minor arc.

Angles formed in a circle joining the endpoints of arcs are called subtended angles. Various types of angles are formed by the arcs, such as angles in the same segment, angles at the circumference or boundary of the circle, angles in a semi-circle, etc.

Arc of a Circle

Objects that are round are called circular objects. We see many such circular objects in our real-life like basketball rings, tires of a bicycle, base of the cylinder, bangles, the cone base, etc. 

The part of the boundary or circumference of the circle is known as an arc of the circle. Thus, an arc is the connected part of the circle. There are two types of arcs formed in the circle, namely major arc and minor arc. The sum of the lengths of the minor arc and the major arc equals the circle’s circumference.

Major Arc – The part of the circle or an arc formed whose length is more than the length of a semi-circle is called a major arc.

In the above figure, the blue colour shaded part of the circle is known as the major arc.

Minor Arc – The part of the circle or an arc formed, whose length is smaller than the length of a semi-circle, is called a minor arc.

In the above figure, the red colour shaded part of the circle is known as the minor arc.

In the above figure, the angle subtended by the minor arc \(P Q\) at \(O\) is \(\angle P O Q\) and the angle subtended by the major arc \(P Q\) at \(O\) is reflex angle \(P O Q\).

Angles Subtended by Equal Arcs at The Centre

If any two chords of a circle are equal, then their corresponding arcs of the circle are the same, and conversely, we can say that if two arcs are the same, then their corresponding chords are equal.

We know that equal chords subtend equal angles at the centre. Hence, we can say that equal chords subtend equal angles at the centre of a circle.

In the above figure, the two equal arcs \(A B\) and \(P D\) angles at the centre \(O\) are equal.
\(\angle A O B=\angle P O D\)

Angle Subtend by an Arc at the Centre

If two line segments originate from the centre of the circle to the endpoints of the arc, then the angle formed between them is called the centre’s subtended angle. We have different theorems related to the relations between angles made by the arc at the centre and angles made at the other part of the circle.

Theorem-1:
The angle subtended by an arc at the centre is double or twice the angle subtended by it at any point on the circle’s remaining part (circumference) at its same side.

Proof:
Consider three different situations, such as the arc is a minor arc, major arc, and the arc is a semi-circle as shown below:

Join \(O\) and \(A\) and extend the ray to \(B\), as shown in the figure.

We know that the exterior angle of the triangle is equal to the sum of the opposite interior angles.

Thus, \(\angle B O Q=\angle O A Q+\angle O Q A……(i)\)

In triangle \(A O Q\),

\(O A=O Q\) (Radius of the circle)

Thus, \(\triangle A O Q\) is an isosceles triangle, and we know that sides opposite to equal angles of the triangle are equal.

Therefore, \(\angle O A Q=\angle O Q A ….. (ii)\)

From \((i)\) and \((ii)\),

\(\angle B O Q=2 \angle O A Q ….. (iii)\)

Similarly, \(\angle B O P=2 \angle O A P……(iv)\)

Adding \((iii)\) and \((iv)\),

\(\angle B O Q+\angle B O P=2 \angle O A Q+2 \angle O A P\)

From the figure,

\(\angle P O Q=2 \angle P A Q\)

Therefore, the angle subtended by the arc at the centre \((\angle P O Q)\) is equal to twice the angle subtended at the circumference \((\angle P A Q)\).

The angle in a Semi-Circle is Right Angle

Consider an arc, which is the semi-circle, as shown in the below figure.

We need to prove, \(\angle P A Q=90^{\circ}\) (Right angle)

Here, an arc \(P Q\) makes a straight angle \(\left(180^{\circ}\right)\) at the centre.

Thus, \(\angle P O Q=180^{\circ}\)

We know that the angle subtended by an arc at the centre is double or twice the angle subtended by it at any point on the circle’s remaining part (circumference).

So, \(\angle P O Q=2 \angle P A Q\)

\(\Rightarrow \angle P A Q=\frac{\angle P O Q}{2}=\frac{180^{\circ}}{2}=90^{\circ}\)

Thus, the angle at the semi-circle is a right angle.

Angles in the Same Segment of a Circle are Equal

The segment is the region formed by the chord and an arc of the circle. There are two segments in a circle, viz. the major segment and the minor segment. Angles formed by the line segments drawn from endpoints of the arc either in the minor segment or the major segment are equal.

Theorem:

Angles in the same segment of a circle are equal. In other words, we can say that an arc subtends equal angles at any part of the circle’s circumference on the same side of the arc.

Proof:

Consider the circle as shown above, in which an arc \(P Q\) subtends angles \(P A Q\) and \(P C Q\) at any two points \(A\) and \(C\) on the circumference of a circle.

We know that the angle subtended by an arc at the centre is double the angle subtended by it at any point on the circle’s remaining part (circumference) on the same side.

Thus, \(\angle P O Q=2 \angle P A Q……(i)\)

Similarly, \(\angle P O Q=2 \angle P C Q…….(ii)\)

From \((i)\) and \((ii)\),

\(2 \angle P A Q=2 \angle P C Q\)

\(\angle P A Q=\angle P C Q\)

Thus, angles made by the arc at the same segments are equal.

Solved Examples on Angle Subtended by an Arc of a Circle

Q.1. \(A, B\) and \(C\) are three points on a circle with centre \(O\) such that \(\angle B O C=30^{\circ}\) and \(\angle A O B =60^{\circ}\). If \(D\) is a point on the circle other than the arc \(\mathrm{ABC}\), find \(\angle \mathrm{ADC}\).

Ans:
Given, \(\angle B O C=30^{\circ}\) and \(\angle A O B=60^{\circ}\)
From the figure,
\(\angle A O C=\angle A O B+\angle B O C\)
\(\Rightarrow \angle A O C=30^{\circ}+60^{\circ}=90^{\circ}\)
Here, \(\angle A O C\) is the angle made by the arc at the centre “\(O\)”, and \(\angle A D C\) is the angle made by the arc at the point \(D\) on the circle.
We know that the angle subtended by an arc at the centre is double or twice the angle subtended by it at any point on the circle’s remaining part (circumference) on the same side.
Thus, \(\angle A O C=2 \angle A D C\)
\(\Rightarrow \angle A D C=\frac{\angle A O C}{2}\)
\(\Rightarrow \angle A D C=\frac{90^{\circ}}{2}=45^{\circ}\)
Hence, the value of \(\angle A D C\) is \(45^{\circ}\)

Q.2. Observe the given figure, in which \(\angle P Q R=100^{\circ}\), where \(P, Q\) and \(R\) are points on a circle with centre \(O\). Find \(\angle O P R\).

Ans:
Given \(\angle P Q R=100^{\circ}\).
We know that the angle subtended by an arc at the centre is double or twice the angle subtended by it at any point on the circle’s remaining part (circumference) at the same side.
Thus, reflex \(\angle P O R=2 \angle P Q R=2 \times 100^{\circ}=200^{\circ}\)
Then, \(\angle P O R=360^{\circ}-\) Reflex \(\angle P O R\)
\(\angle P O R=360^{\circ}-200^{\circ}=160^{\circ}\)
In triangle \(O P R\),
\(O P=O R\) (Radius of the circle)
We know that sides opposite to equal angles are equal.
\(\angle O P R=\angle O R P\)
In a triangle \(O P R\), by angle sum property,
\(\angle O P R+\angle O R P+\angle P O R=180^{\circ}\)
\(\Longrightarrow 2 \angle O P R+160^{\circ}=180^{\circ}\)
\(\Longrightarrow 2 \angle O P R=180^{\circ}-160^{\circ}=20^{\circ}\)
\(\Longrightarrow \angle O P R=\frac{20^{\circ}}{2}=10^{\circ}\)
Therefore the value of \(\angle O P R=10^{\circ}\).

Q.3. Observe the given figure, in which \(\angle A B C=69^{\circ}, \angle A C B=31^{\circ}\), find \(\angle B D C\).

Ans:
Given, \(\angle A B C=69^{\circ}, \angle A C B=31^{\circ}\).
We know that angles made by the arc at the same segments are equal.
Thus, \(\angle B D C=\angle B A C…..(i)\)
In the triangle \(A B C\), by angle sum property
\(\Longrightarrow \angle B A C+\angle A B C+\angle A C B=180^{\circ}\)
\(\Longrightarrow \angle B A C+69^{\circ}+31^{\circ}=180^{\circ}\)
\(\Longrightarrow \angle B A C+100^{\circ}=180^{\circ}\)
\(\Longrightarrow \angle B A C=180^{\circ}-100^{\circ}=80^{\circ}\)
From \((i)\),
\(\angle B D C=80^{\circ}\)

Q.4. \(P Q\) is the diameter of the circle. Find the value of \(\angle P A Q\) from the given figure.

Ans:
Here, an arc \(P Q\) makes a straight angle \(\left(180^{\circ}\right)\) at the centre.
Thus, \(\angle P O Q=180^{\circ}\)
We know that the angle subtended by an arc at the centre is double or twice the angle subtended by it at any point on the circle’s remaining part (circumference) on the same side.
So, \(\angle P O Q=2 \angle P A Q\)
\(\Rightarrow \angle P A Q=\angle \frac{P O Q}{2}=\frac{180^{\circ}}{2}=90^{\circ}\)
Therefore, \(\angle P A Q=90^{\circ}\).

Q.5. Prove that angles \(P A Q\) and \(P C Q\) are equal in the given figure.

Ans:
We know that the angle subtended by an arc at the centre is double or twice the angle subtended by it at any point on the circle’s remaining part (circumference).
Thus, \(\angle P O Q=2 \angle P A Q…….(i)\)
Similarly, \(\angle P O Q=2 \angle P C Q……(ii)\)
From \((i)\) and \((ii)\),
\(2 \angle P A Q=2 \angle P C Q\)
\(\angle P A Q=\angle P C Q\)
Hence, proved.

Summary

In this article, we have studied the definitions of the circle, arcs, minor and major arcs. We also studied the angles subtended by the arc.

This article describes few theorems related to angles made by the arcs at the centre and at any other part of the circle.  We also described some solved examples and frequently asked questions.

FAQs on Angle Subtended by an Arc of a Circle

Q.1. What is the angle in a semi-circle?
Ans: The angle in a semi-circle is the right angle.

Q.2. What is an arc of a circle?
Ans: The part of the boundary or circumference of the circle is known as an arc of the circle.

Q.3. Are angles at the same segments equal?
Ans: Yes. The angles at the same segment formed by the arc of a circle are equal.

Q.4. How do you find the angle subtended by the arc at the centre?
Ans: The angle subtended by the arc at the centre of a circle is twice the angle subtended at any part of the circle on the same side.

Q.5. What is the relationship between an inscribed angle and the subtended arc?
Ans: The inscribed angle is half the angle made at the centre by the subtended arc.