What is Area of an Annulus?

S3-SA2-0151

Grade Level:

Class 7

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

Definition

What is it?

An annulus is a ring-shaped region, like a flat donut. The area of an annulus is the space it covers, calculated by subtracting the area of the smaller inner circle from the area of the larger outer circle.

Simple Example

Quick Example

Imagine you have a big circular roti and you cut out a smaller circular piece from its center to make a ring. The flat area of this roti ring is the area of an annulus. It's the 'roti' part that's left.

Worked Example

Step-by-Step

Let's find the area of an annulus where the outer circle has a radius of 7 cm and the inner circle has a radius of 3 cm. (Use pi = 22/7) --- Step 1: Find the area of the outer circle. Area_outer = pi * R^2 = (22/7) * (7)^2 = (22/7) * 49 = 22 * 7 = 154 sq cm. --- Step 2: Find the area of the inner circle. Area_inner = pi * r^2 = (22/7) * (3)^2 = (22/7) * 9 = 198/7 = 28.28 sq cm (approx). --- Step 3: Subtract the inner circle's area from the outer circle's area. Area_annulus = Area_outer - Area_inner = 154 - 28.28 = 125.72 sq cm. --- Answer: The area of the annulus is approximately 125.72 sq cm.

Why It Matters

Understanding annulus area helps engineers design pipes and washers, and architects plan circular structures with open centers. This concept is used in fields like mechanical engineering for designing gears and in computer graphics for creating ring-like shapes.

Common Mistakes

MISTAKE: Adding the areas of the two circles instead of subtracting. | CORRECTION: An annulus is the *space between* two circles, so you must subtract the smaller area from the larger area.

MISTAKE: Using the same radius for both circles. | CORRECTION: Always use the larger radius (R) for the outer circle and the smaller radius (r) for the inner circle.

MISTAKE: Forgetting to square the radius when calculating the area of each circle. | CORRECTION: Remember the formula for the area of a circle is pi * radius^2, not pi * radius.

Practice Questions

Try It Yourself

QUESTION: A circular park has an outer radius of 10 meters. A circular pond inside it has a radius of 4 meters. What is the area of the walking path around the pond? (Use pi = 3.14) | ANSWER: 263.76 sq meters

QUESTION: If an annulus has an outer diameter of 20 cm and an inner diameter of 12 cm, what is its area? (Hint: Diameter = 2 * Radius, use pi = 3.14) | ANSWER: 200.96 sq cm

QUESTION: A circular disc has a radius of 14 cm. A smaller circular hole is punched out from its center, creating an annulus with an area of 528 sq cm. What is the radius of the punched-out hole? (Use pi = 22/7) | ANSWER: 10 cm

MCQ

Quick Quiz

Which formula correctly represents the area of an annulus?

pi * (R + r)

pi * (R^2 + r^2)

pi * (R^2 - r^2)

2 * pi * (R - r)

The Correct Answer Is:

The area of an annulus is found by subtracting the area of the inner circle (pi * r^2) from the area of the outer circle (pi * R^2), which simplifies to pi * (R^2 - r^2).

Real World Connection

In the Real World

In India, think of a 'churidar' bangle or a 'diya' stand with a hollow center. The flat surface of such items, excluding the hole, represents an annulus. Engineers at ISRO might use this concept when designing components for rocket engines that have central openings.

Key Vocabulary

Key Terms

ANNULUS: A ring-shaped region between two concentric circles | RADIUS: The distance from the center of a circle to its edge | DIAMETER: The distance across a circle passing through its center (twice the radius) | CONCENTRIC CIRCLES: Circles that share the same center point

What's Next

What to Learn Next

Great job learning about the area of an annulus! Next, you can explore the 'Surface Area of 3D Shapes' like cylinders and cones. This will help you understand how to calculate areas for objects that are not flat.