Annulus

The term 'annulus' comes from the Latin word annuli, meaning 'little ring.' An annulus is a flat, plane figure formed between two concentric circles, creating a ring-like shape. The area of an annulus refers to the region enclosed between these two concentric circles, which lie in the same plane.

In the figure below, the green region represents the annulus.

It is a two-dimensional flat figure with a circular form, consisting of the space between two concentric circles that share the same center. It can be thought of as a circular disk with a hole in the middle.

Some real-life examples of an Annulus Shape are:

Doughnuts,
Finger rings,
CD,
Shape of a dartboard

Formulas of Annulus

The formulas for the annulus are based on the radii of the two concentric circles. Let the radius of the outer circle be R, and the radius of the inner circle be r.

Here are the key formulas related to an annulus:

Area of Annulus

The area of the annulus can be calculated by finding the area of the outer circle and the inner circle, then we have to subtracting the area of both circles to get the result.

In the above figure, two circles have having common centre O. The shaded portion in green indicates an annulus. To find the area of the annulus, we are required to find the area of both circles. Therefore,

Area of Annulus = π(R² - r²)

Where:

R is the radius of the outer circle,
r is the radius of the inner circle.

In this formula, the area of the outer circle (πR²) is calculated, and the area of the inner circle (πr²) is subtracted.

Read More: Annulus Area Formula

Perimeter of Annulus

The Perimeter (or circumference) of the annulus involves the sum of the circumferences of the inner and outer circles. Since the annulus is a ring, its perimeter is the sum of the circumferences of the two concentric circles.

Perimeter of Annulus = 2π(R + r)

Where:

R is the radius of the outer circle,
r is the radius of the inner circle.

This formula gives the total perimeter of the annular region by adding the circumferences of both circles.

These formulas are useful for calculating the area and perimeter of an annulus based on the radii of the two concentric circles.

Solved Question On Annulus

Question 1: Calculate the area of the annulus if the outer radius is 12 units and the inner radius is 7 units.

Solution:

Given ,
Outer radius(R) = 12 units
Inner radius (r) = 7 units
Area of outer circle = πR² = 3.142 * 12 * 7 = 263.928 units
Area of inner circle = πr² = 3.142 * 7 * 7 = 153.958 units
The formula to find the area of annulus = (area of outer circle - area of inner circle ) square units
Area = (263.928 - 153.958) square units = 109.97 square units
Therefore, the required area is 109.97 square units.

Question 2: If the outer radius of the ring is 9 units and the inner radius of the ring is 4 units, what would be the perimeter of the ring?

Solution:

Given,
Outer radius(R) = 9 units
Inner radius(r) = 4 units
we already know that perimeter of annulus = 2π(R + r)
Perimeter = 2 * 3.142 (9 + 4)
Perimeter = 81.692 units
Therefore ,the perimeter of the ring is 81.692 units .

Question 3: A circular disc has a hole cut out from its center, forming an annulus. The outer diameter of the disc is 20 units, and the inner diameter (the hole) is 16 cm. Find the area of the annulus.

Solution:

Given,
Outer diameter = 20 units
Outer radius = 20/2 = 10 units
Inner diameter = 16 units
Inner radius = 16/2 = 8 units
Area of Annulus = π(R² -r²) square units
Area = 3.142( 10² - 8²)
Area = 3.142(100 - 64)
Area = 3.142(36)
Area = 113.112 units²
Therefore, the area of the annulus is 113.112 units²

Question 4: A circular ring has an outer radius of 15 cm and an area of 660 cm².Find the inner radius.

Solution:

Given,
Outer radius = 15 cm
Area of circular ring = 660cm²
Inner radius = r cm
Need to find value of r
Area = π(R² - r²) square units
660 = π(15²- r²)
660 = π(225 - r²)
660 = 3.142 (225 -r²)
660/3.142 = 225 - r²
210 = 225 - r²
r² = 225 - 210
r² = 15
r = √15
r = 3.87 cm
Therefore, the value of inner radius is 3.87 cm

Unsolved Practice Question of Annulus

Question 1: Calculate the area of the annulus if the outer radius is 15 units and the inner radius is 8 units.

Question 2: A steel pipe has an outer radius of 80 units and an inner radius of 60 units. What are area of the cross-section?

Question 3: If the outer radius is 8 units and the inner radius of a ring is 5 units, what would be the perimeter of the ring?

Question 4: If the area of an annulus is 1092 inches and its width is 3 cm, find the radii of the inner and outer circles. (Hint Width = Outer radius - Inner radius).

Answer sheet
1) 505.862 units²
2) 2199.4 units²
3) 81.68 units
4)Inner radius = 56.41 inches
Outer radius = 59.41 inches

Circle
Sector of a Circle
Geometry
Area and Perimeter of Shapes

Formulas of Annulus

Area of Annulus

Perimeter of Annulus

Solved Question On Annulus

Unsolved Practice Question of Annulus

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