What is the Area of a Ring (Annulus)?

S3-SA2-0302

Grade Level:

Class 6

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

Definition

What is it?

The area of a ring, also called an annulus, is the space covered by a flat shape that looks like a circular ring. Imagine a bigger circle with a smaller circle cut out from its center. The area of the leftover part is the area of the ring.

Simple Example

Quick Example

Think of a 'CD' or a 'DVD'. It's a big circle with a smaller circular hole in the middle. The shiny part where the data is stored (not the hole) is an example of an annulus. Its area is the area of the whole CD minus the area of the hole.

Worked Example

Step-by-Step

Let's find the area of a bangle. The outer radius (from the center to the outer edge) is 7 cm. The inner radius (from the center to the inner edge) is 5 cm. (Use pi = 22/7)

Step 1: Identify the radii. Outer radius (R) = 7 cm, Inner radius (r) = 5 cm.
---Step 2: Calculate the area of the outer circle. Area_outer = pi * R * R = (22/7) * 7 * 7 = 22 * 7 = 154 sq cm.
---Step 3: Calculate the area of the inner circle. Area_inner = pi * r * r = (22/7) * 5 * 5 = (22 * 25) / 7 = 550 / 7 = 78.57 sq cm (approx).
---Step 4: Subtract the inner area from the outer area to get the ring's area. Area_ring = Area_outer - Area_inner = 154 - 78.57 = 75.43 sq cm.
---The area of the bangle (ring) is approximately 75.43 square centimeters.

Why It Matters

Understanding annulus area is crucial in engineering for designing pipes, washers, and circular components. In physics, it helps calculate forces on ring-shaped objects. It's also used in computer graphics for rendering circular objects with holes and in architecture for designing circular pathways or structures.

Common Mistakes

MISTAKE: Adding the areas of the two circles instead of subtracting. | CORRECTION: The ring is what's LEFT after cutting out the inner circle, so you must subtract the inner area from the outer area.

MISTAKE: Using the diameter instead of the radius in the area formula (pi * r * r). | CORRECTION: Remember, the formula uses 'r' for radius, which is half of the diameter. If given diameter, divide by 2 first.

MISTAKE: Mixing up the inner and outer radii when subtracting. | CORRECTION: Always subtract the area of the SMALLER circle (inner radius) from the area of the LARGER circle (outer radius).

Practice Questions

Try It Yourself

QUESTION: A circular park has an outer radius of 10 meters. A circular pond in its center has a radius of 3 meters. What is the area of the grass path around the pond? (Use pi = 3.14) | ANSWER: Area_outer = 3.14 * 10 * 10 = 314 sq m. Area_inner = 3.14 * 3 * 3 = 28.26 sq m. Area_path = 314 - 28.26 = 285.74 sq m.

QUESTION: A circular disc has an outer diameter of 14 cm and a circular hole of diameter 6 cm in its center. Find the area of the remaining disc. (Use pi = 22/7) | ANSWER: Outer radius (R) = 14/2 = 7 cm. Inner radius (r) = 6/2 = 3 cm. Area_outer = (22/7) * 7 * 7 = 154 sq cm. Area_inner = (22/7) * 3 * 3 = 198/7 = 28.29 sq cm (approx). Area_disc = 154 - 28.29 = 125.71 sq cm.

QUESTION: A circular mat has an outer radius of 20 cm. A border of 2 cm width runs along its inner edge. What is the area of the border? (Use pi = 3.14) | ANSWER: Outer radius (R) = 20 cm. Inner radius (r) = Outer radius - border width = 20 - 2 = 18 cm. Area_outer = 3.14 * 20 * 20 = 1256 sq cm. Area_inner = 3.14 * 18 * 18 = 1017.36 sq cm. Area_border = 1256 - 1017.36 = 238.64 sq cm.

MCQ

Quick Quiz

Which formula correctly represents the area of a ring (annulus) where R is the outer radius and r is the inner radius?

pi * (R + r)

pi * (R * R + r * r)

pi * (R * R - r * r)

pi * R * R / r * r

The Correct Answer Is:

The area of a ring is found by subtracting the area of the inner circle from the area of the outer circle. The area of a circle is pi * radius * radius. So, it's (pi * R * R) - (pi * r * r), which simplifies to pi * (R * R - r * r).

Real World Connection

In the Real World

Imagine the circular 'donuts' or 'washers' used in plumbing to prevent leaks, or the circular tracks in an athletics stadium. Even the design of some ISRO satellite components involves calculating the area of ring-shaped parts for optimal weight distribution and material usage.

Key Vocabulary

Key Terms

ANNULUS: Another name for a ring-shaped object | RADIUS: The distance from the center of a circle to its edge | DIAMETER: The distance across a circle through its center (twice the radius) | PI (pi): A special number (approx 3.14 or 22/7) used in circle calculations | AREA: The amount of surface covered by a flat shape

What's Next

What to Learn Next

Great job learning about the area of a ring! Next, you can explore the 'Circumference of a Circle' to understand the distance around a circular object. Then, you can move on to 'Area of Composite Shapes' which combines different basic shapes, including rings, to find their total area.