Area of a Ring
2026-02-28 17:24 Diff
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Last updated on September 17, 2025

Area is the space inside the boundaries of a two-dimensional shape or surface. There are different formulas for finding the area of various shapes/figures. These are widely used in architecture and design. In this section, we will find the area of a ring.

What is the Area of a Ring?

A ring is a circular band that is formed between two concentric circles, meaning circles that share the same center.

The area of the ring is the space enclosed between the outer circle and the inner circle.

Area of the Ring Formula

To find the area of the ring, we use the formula: π(R² - r²), where R is the radius of the outer circle and r is the radius of the inner circle. Now let’s see how the formula is derived.

Derivation of the formula: The area of the ring is the difference between the area of the larger circle and the area of the smaller circle. The area of the larger circle with radius R is πR². The area of the smaller circle with radius r is πr². Thus, the area of the ring is π(R² - r²).

How to Find the Area of a Ring?

We can find the area of a ring using the difference between the areas of two circles. The formula to find the area is: If the radii R and r are given, we find the area of the ring using the formula Area = π(R² - r²).

For example, if R is 10 cm and r is 6 cm, what will be the area of the ring? Area = π(R² - r²) = π(10² - 6²) = π(100 - 36) = 64π The area of the ring is 64π cm² or approximately 201.06 cm² when using π ≈ 3.14.

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Unit of Area of a Ring

We measure the area of a ring in square units. The measurement depends on the system used:

In the metric system, the area is measured in square meters (m²), square centimeters (cm²), and square millimeters (mm²).

In the imperial system, the area is measured in square inches (in²), square feet (ft²), and square yards (yd²).

Special Cases or Variations for the Area of a Ring

There are not many variations in finding the area of a ring, as it primarily depends on the radii of the two concentric circles. However, consider these:

Case 1: When the radii are given Use the formula Area = π(R² - r²).

Case 2: When the diameters are given Convert diameters into radii by dividing each by 2, then use the formula Area = π(R² - r²).

Tips and Tricks for Area of a Ring

To ensure correct results while calculating the area of a ring, consider the following tips and tricks:

  • The radii of both circles must be measured from the same center point.
     
  • Make sure to square the radii before subtracting. Use consistent units when calculating the area.
     
  • Ensure the outer radius is greater than the inner radius.

Common Mistakes and How to Avoid Them in Area of a Ring

It is common for students to make mistakes while finding the area of a ring. Let’s take a look at some common mistakes.

Problem 1

The radii of two concentric circles are 8 m and 5 m. What will be the area of the ring?

Okay, lets begin

We will find the area as 39π m².

Explanation

Here, the outer radius R is 8 m and the inner radius r is 5 m.

The area of the ring = π(R² - r²) = π(8² - 5²) = π(64 - 25) = 39π m².

Well explained 👍

Problem 2

What will be the area of the ring if the outer diameter is 20 cm and the inner diameter is 10 cm?

Okay, lets begin

We will find the area as 75π cm².

Explanation

First, convert the diameters to radii:

R = 20/2 = 10 cm, r = 10/2 = 5 cm.

Then, use the formula: Area = π(R² - r²) = π(10² - 5²) = π(100 - 25) = 75π cm².

Well explained 👍

Problem 3

The area of a ring is 100π m², and the outer radius is 15 m. What is the inner radius?

Okay, lets begin

We find the inner radius as 5 m.

Explanation

Use the formula: Area = π(R² - r²).

Here, the area is 100π m², and R is 15 m.

100π = π(15² - r²) 100 = 225 - r² r² = 225 - 100 = 125 r = √125 = 5√5 ≈ 11.18 m.

Well explained 👍

Problem 4

Find the area of the ring if the outer radius is 12 cm and the inner radius is 7 cm.

Okay, lets begin

We will find the area as 119π cm².

Explanation

Use the formula: Area = π(R² - r²).

Substitute the values: Area = π(12² - 7²) = π(144 - 49) = 95π cm².

Well explained 👍

Problem 5

Help Emma find the area of the ring if the outer radius is 9 m and the inner radius is 3 m.

Okay, lets begin

We will find the area as 72π m².

Explanation

Given R = 9 m and r = 3 m, use the formula: Area = π(R² - r²) = π(9² - 3²) = π(81 - 9) = 72π m².

Well explained 👍

FAQs on Area of a Ring

1.Is it possible for the area of the ring to be negative?

No, the area of a ring can never be negative. The area of any shape will always be positive.

2.How to find the area of a ring if the diameters are given?

Convert the diameters to radii by dividing by 2, then use the formula Area = π(R² - r²).

3.How to find the area of the ring if only the outer radius is given?

You cannot find the area of the ring with only the outer radius. You need both the outer and inner radii.

4.How is the circumference of the ring calculated?

The circumference of a ring is not typically calculated as a whole. Instead, calculate the circumferences of both circles separately using 2πR and 2πr.

5.What is meant by the area of a ring?

The area of the ring is the total space enclosed between the two concentric circles.

Seyed Ali Fathima S

About the Author

Seyed Ali Fathima S a math expert with nearly 5 years of experience as a math teacher. From an engineer to a math teacher, shows her passion for math and teaching. She is a calculator queen, who loves tables and she turns tables to puzzles and songs.

Fun Fact

: She has songs for each table which helps her to remember the tables