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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.
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.
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).
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.
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).
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).
To ensure correct results while calculating the area of a ring, consider the following tips and tricks:
It is common for students to make mistakes while finding the area of a ring. Let’s take a look at some common mistakes.
The radii of two concentric circles are 8 m and 5 m. What will be the area of the ring?
We will find the area as 39π m².
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.
What will be the area of the ring if the outer diameter is 20 cm and the inner diameter is 10 cm?
We will find the area as 75π cm².
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.
The area of a ring is 100π m², and the outer radius is 15 m. What is the inner radius?
We find the inner radius as 5 m.
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.
Find the area of the ring if the outer radius is 12 cm and the inner radius is 7 cm.
We will find the area as 119π cm².
Use the formula: Area = π(R² - r²).
Substitute the values: Area = π(12² - 7²) = π(144 - 49) = 95π cm².
Help Emma find the area of the ring if the outer radius is 9 m and the inner radius is 3 m.
We will find the area as 72π m².
Given R = 9 m and r = 3 m, use the formula: Area = π(R² - r²) = π(9² - 3²) = π(81 - 9) = 72π m².
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.
: She has songs for each table which helps her to remember the tables