What is the Angle Subtended by an Arc at the Centre?

S3-SA2-0544

Grade Level:

Class 6

AI/ML, Data Science, Physics, Economics, Cryptography, Computer Science, Engineering

Definition

What is it?

The angle subtended by an arc at the centre is the angle formed when you connect the two ends of the arc to the centre point of the circle. Imagine a slice of pizza; the angle at the tip of the slice (which is the centre of the whole pizza) is the angle subtended by the crust (the arc).

Simple Example

Quick Example

Think of a clock. The minute hand moves from 12 to 3. The path it covers on the edge of the clock is an arc. If you draw lines from '12' and '3' to the very centre of the clock, the angle formed at the centre is the angle subtended by that arc. It would be 90 degrees.

Worked Example

Step-by-Step

Let's say a circle has a total angle of 360 degrees. An arc covers one-fourth (1/4) of the circle's total circumference.

STEP 1: Understand the total angle in a circle. A full circle is 360 degrees.
---STEP 2: Identify the fraction of the circle the arc covers. Here, it's 1/4.
---STEP 3: To find the angle subtended at the centre, multiply the total angle by this fraction.
---STEP 4: Calculation: Angle = (1/4) * 360 degrees.
---STEP 5: Angle = 90 degrees.

So, the angle subtended by an arc that is 1/4 of the circle is 90 degrees.

Why It Matters

Understanding angles in circles is super important for many cool fields. In Computer Science, it helps create graphics and animations for games and apps. In Engineering, it's used to design gears and circular structures like flyovers and stadiums. Knowing this helps build a strong base for future innovations!

Common Mistakes

MISTAKE: Confusing the angle subtended at the centre with the angle subtended at any other point on the circle. | CORRECTION: The angle at the centre is unique and directly related to the arc length. An angle subtended at any point on the circumference (not the centre) will be half of the angle at the centre for the same arc.

MISTAKE: Not connecting the ends of the arc to the *exact* centre of the circle. | CORRECTION: Always draw lines from the two endpoints of the arc straight to the centre of the circle. The angle is formed precisely at this centre point.

MISTAKE: Assuming all arcs subtend the same angle. | CORRECTION: The angle subtended by an arc at the centre depends on the length of the arc. A longer arc will subtend a larger angle, and a shorter arc will subtend a smaller angle.

Practice Questions

Try It Yourself

QUESTION: An arc covers half of a circle. What angle does it subtend at the centre? | ANSWER: 180 degrees

QUESTION: If an arc subtends an angle of 60 degrees at the centre, what fraction of the entire circle's circumference does it cover? | ANSWER: 1/6 (because 60/360 = 1/6)

QUESTION: A circular park has a walking track. A segment of the track, which is 1/8th of the total track length, is used for a special event. What angle does this segment subtend at the centre of the park? | ANSWER: 45 degrees (1/8 * 360 = 45)

MCQ

Quick Quiz

What is the angle subtended at the centre by an arc that covers one-third of a circle?

90 degrees

120 degrees

180 degrees

60 degrees

The Correct Answer Is:

A full circle is 360 degrees. If an arc covers one-third (1/3) of the circle, then the angle subtended at the centre is (1/3) * 360 degrees, which equals 120 degrees.

Real World Connection

In the Real World

When you order a pizza, the angle of each slice at the centre is the angle subtended by its crust (arc). In satellite communication, engineers calculate the angle subtended by the Earth's surface at the satellite to determine how much area can be covered by a signal. Even in making a 'rangoli', understanding these angles helps create perfectly symmetrical designs.

Key Vocabulary

Key Terms

ARC: A part of the circumference of a circle | CENTRE: The middle point of a circle, equidistant from all points on its circumference | SUBTEND: To form an angle at a particular point | CIRCUMFERENCE: The perimeter or boundary of a circle | ANGLE: The space (usually measured in degrees) between two intersecting lines or surfaces at or close to the point where they meet.

What's Next

What to Learn Next

Great job understanding this! Next, you can explore 'Relationship between Angle at Centre and Angle at Circumference'. This will show you how the angle you just learned about connects to angles formed on the edge of the circle, making your geometry knowledge even stronger!