What is the Area of an Annulus?

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Grade Level:

Class 6

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

Definition

What is it?

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

Simple Example

Quick Example

Imagine a big circular rangoli design with a smaller circular design painted inside it. The coloured area between the two circles is an annulus. To find how much space that coloured part takes, you'd calculate the area of the big rangoli and subtract the area of the small rangoli.

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 * 7 = 22 * 7 = 154 sq cm.
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Step 2: Find the area of the inner circle. Area_inner = pi * r^2 = (22/7) * 3 * 3 = 22 * 9 / 7 = 198 / 7 = 28.28 sq cm (approx).
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Step 3: Subtract the inner area from the outer area. Area_annulus = Area_outer - Area_inner = 154 - 28.28 = 125.72 sq cm.
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Answer: The area of the annulus is approximately 125.72 square centimeters.

Why It Matters

Understanding annulus area helps engineers design pipes, washers, and circular components in machines. It's also used in computer graphics for creating ring-like shapes and in physics to calculate forces on circular objects. From building bridges to making cool video games, this concept is super useful!

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 always subtract the area of the smaller circle from the area of the larger circle.

MISTAKE: Using the diameter instead of the radius in the area formula (pi * r^2). | CORRECTION: Remember, 'r' stands for radius. If given the diameter, divide it by 2 to get the radius before using the formula.

MISTAKE: Forgetting to use the correct value of pi (22/7 or 3.14) or rounding too early. | CORRECTION: Always use the value of pi specified in the question, or 22/7 for calculations involving multiples of 7, and 3.14 otherwise. Round only at the final step.

Practice Questions

Try It Yourself

QUESTION: A circular swimming pool has a diameter of 10 meters. A circular pathway around it has an outer diameter of 12 meters. What is the area of the pathway? (Use pi = 3.14) | ANSWER: Outer radius = 6m, Inner radius = 5m. Area_outer = 3.14 * 6^2 = 113.04 sq m. Area_inner = 3.14 * 5^2 = 78.5 sq m. Area_annulus = 113.04 - 78.5 = 34.54 sq m.

QUESTION: A circular disc has a radius of 14 cm. A smaller circular hole with a radius of 7 cm is cut out from its center. What is the area of the remaining disc? (Use pi = 22/7) | ANSWER: Outer radius = 14 cm, Inner radius = 7 cm. Area_outer = (22/7) * 14 * 14 = 616 sq cm. Area_inner = (22/7) * 7 * 7 = 154 sq cm. Area_annulus = 616 - 154 = 462 sq cm.

QUESTION: A circular coin has an inner radius of 1.5 cm and an outer radius of 2 cm. If 5 such coins are arranged side-by-side, what is the total area of the annulus part of all 5 coins? (Use pi = 3.14) | ANSWER: Area of one annulus = pi * (R^2 - r^2) = 3.14 * (2^2 - 1.5^2) = 3.14 * (4 - 2.25) = 3.14 * 1.75 = 5.495 sq cm. Total area for 5 coins = 5 * 5.495 = 27.475 sq cm.

MCQ

Quick Quiz

What is the formula for the area of an annulus?

pi * (R + r)

pi * (R^2 + r^2)

pi * (R^2 - r^2)

2 * pi * R - 2 * pi * 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

Think about the rubber 'O-rings' used in water taps or pressure cookers to prevent leaks. These are perfect examples of annuli! Engineers need to calculate their area to ensure they fit perfectly and provide a good seal. Even the circular pathways in a park around a statue are annuli.

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. | Concentric Circles: Circles that share the same center point. | Area: The amount of surface covered by a two-dimensional shape. | Pi (pi): A mathematical constant, approximately 3.14 or 22/7, used in circle calculations.

What's Next

What to Learn Next

Great job learning about the area of an annulus! Next, you can explore the perimeter (circumference) of circles and how to find the area of sectors and segments of a circle. This will help you understand more complex shapes and their properties.