What is the Angle Subtended by an Arc?

S6-SA2-0318

Grade Level:

Class 10

AI/ML, Physics, Biotechnology, Space Technology, Chemistry, Engineering, Medicine

Definition

What is it?

The angle subtended by an arc is the angle formed by drawing lines from the two endpoints of the arc to a specific point. This point can be either the center of the circle or any point on the circumference. It essentially measures how 'wide' the arc appears from that particular point.

Simple Example

Quick Example

Imagine you are looking at a slice of pizza. The crust of the pizza forms an arc. If you draw lines from the ends of that crust slice to the center of the pizza, the angle formed at the center is the angle subtended by that crust arc at the center. If you instead imagine looking at the same crust from a point on the pizza's edge, the angle you see would be the angle subtended at the circumference.

Worked Example

Step-by-Step

PROBLEM: A circular park has a walking track. An arc of this track measures 60 degrees at the center of the park. What angle does this arc subtend at any point on the remaining part of the circumference?

STEP 1: Understand the given information. The angle subtended by the arc at the center (let's call it Angle_C) is 60 degrees.
---STEP 2: Recall the theorem that relates the angle at the center to the angle at the circumference. The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circumference.
---STEP 3: Let the angle subtended by the arc at the circumference be Angle_P.
---STEP 4: According to the theorem, Angle_C = 2 * Angle_P.
---STEP 5: Substitute the given value: 60 degrees = 2 * Angle_P.
---STEP 6: Solve for Angle_P: Angle_P = 60 degrees / 2.
---STEP 7: Calculate the result: Angle_P = 30 degrees.

ANSWER: The arc subtends an angle of 30 degrees at any point on the remaining part of the circumference.

Why It Matters

Understanding angles subtended by arcs is crucial in fields like engineering for designing curved structures and in space technology for calculating satellite orbits. It helps engineers design safe bridges and architects create beautiful, stable domes. This concept is also used in computer graphics to render realistic curved shapes.

Common Mistakes

MISTAKE: Confusing the angle subtended at the center with the angle subtended at the circumference. | CORRECTION: Always remember the angle at the center is twice the angle at the circumference for the same arc.

MISTAKE: Assuming the angle subtended at the circumference is always half, even if the point is on the minor arc. | CORRECTION: The theorem applies to a point on the *remaining part* of the circumference, meaning the major arc if the angle at center is for the minor arc.

MISTAKE: Not identifying the correct arc when multiple arcs are present in a diagram. | CORRECTION: Clearly mark the two endpoints of the arc in question and trace the lines from these endpoints to the point where the angle is being measured.

Practice Questions

Try It Yourself

QUESTION: An arc in a circle subtends an angle of 90 degrees at the center. What is the angle it subtends at a point on the circumference? | ANSWER: 45 degrees

QUESTION: If an arc subtends an angle of 70 degrees at a point on the circumference, what angle does it subtend at the center of the circle? | ANSWER: 140 degrees

QUESTION: A circular kolam design has a curved segment that makes a 120-degree angle at the center. If you are standing on the edge of the kolam, what angle would this segment appear to make? | ANSWER: 60 degrees

MCQ

Quick Quiz

An arc of a circle makes an angle of 80 degrees at the center. What angle does it make at any point on the remaining part of the circumference?

160 degrees

80 degrees

40 degrees

20 degrees

The Correct Answer Is:

The angle subtended by an arc at the center is double the angle subtended by the same arc at any point on the remaining part of the circumference. So, 80 degrees / 2 = 40 degrees.

Real World Connection

In the Real World

When ISRO launches a satellite, engineers need to precisely calculate its orbital path around Earth. The Earth's curvature can be thought of as a very large arc. Understanding the angles subtended by different parts of this 'arc' at the satellite's position helps them ensure the satellite stays in the correct orbit and can communicate effectively with ground stations.

Key Vocabulary

Key Terms

ARC: A part of the circumference of a circle. | CIRCUMFERENCE: The boundary or perimeter of a circle. | SUBTEND: To form an angle at a particular point. | CENTER: The middle point of a circle, equidistant from all points on the circumference.

What's Next

What to Learn Next

Great job learning about angles subtended by arcs! Next, you should explore 'Cyclic Quadrilaterals' and 'Tangents to a Circle'. These concepts build directly on your understanding of arcs and angles, helping you solve more complex geometry problems and understand real-world shapes even better.