Annulus Area Formula

The area of an annulus is the difference between the areas of the two circles of radius R and r.

• R is the radius of the bigger circle, while r is the radius of the smaller circle.
• The common area between the two circles is known as the annulus region, and the formula used to find that area is known as the annulus formula.
• It can also be termed as a circle with a hole in the center of radius r. 

The shaded area representing the annulus formula is shown below:

• The radius of the inner circle = ‘r’
• Area of the inner circle = πr²
• The radius of the outer circle = ‘R’
• Area of the outer circle = πR²

The difference (area of annulus) is generally calculated by subtracting a small value from the larger one.

The annulus formula is given by:

π (R² – r²)
Or
π (R – r)(R + r)

For example: Find the area of an annulus with outer radius R = 5cm and inner radius r = 3cm.

Solution:

Annulus formula will be π (R² – r²)
= π (5² – 3²)
= π (25 – 9)
= π (16)

Solved Question on Annulus Area Formula

Question 1: Find the area of the circular racing track formed between the circular field of radius 600m and 300m.

Solution: 

The area of the circular track between the two tracks can be calculated using the annulus formula given by: π (R² – r²)

Where,
R = 600
r = 300

= π (600² – 300²)
= π (360000 – 90000)
= π (270000)
= 847,800 m²

Thus, the area of the circular track is 847,800 square meter.  

Question 2: Find the area of the circular racing track formed between the two circles of radius 250m and 400m.

Solution: 

The area of the circular track between the two tracks can be calculated using the annulus formula given by: π (R² – r²)

Where,
R = 400
r = 250

R is the radius of outer circle and ‘r’ is the radius of inner circle.  
= π (400² – 250²)
= π (160000 – 62500)
= π (97500)
= 306150 m²

Thus, the area of the circular track is 306150 square meter.  

Question 3: Find the area of a path 12cm wide surrounded by a lawn of diameter 300cm.

Solution: 

Given:
Width of the path = 12cm
Diameter of the circular lawn = 300cm
Radius of the inner lawn = diameter/2  
= 300/2
= 150cm

Radius of outer circle = radius of inner circle + width of the path
= 150 + 12
= 162cm

The annulus formula given by: π (R² – r²)

Where,
R = 162
r = 150
A = π (R - r)(R + r)

A = 3.1415 (162 – 150)(162 + 150)
A = 3.1415 ×12 × 312
A = 11761.776 cm²

Thus, the area of the path is 11761.776 square centimeter.  

Question 4: Find the radii of the inner and outer circles if the annulus area and the width of the circular field are 3600m² and 10m?

Solution: 

Given: A = 3600m²
Width = 10m
Width = R – r

Where,
R = radius of outer circle
r = radius of inner circle
R – r = 10
R = 10 + r

The annulus formula is given by: A = π (R – r)(R + r)

Substituting the value of R in the annulus formula, we get:
A = π (10 + r – r)(10 + r + r)
A = π (10)(10 + 2r)

The area of the width or the annulus area is 3600m². Substituting in the above equation, we get:

3600 = π (10)(10 + 2r)
3600 = π (100 + 20r)
100 + 20r = 1145.9
20r = 1145.9 – 100
20r = 1045.9
r = 52.3m

The radius of outer circle = r + 10
= 52.3 + 10
= 62.3m

Thus, the radii of the inner and outer circle are 52.3m and 62.3m.

Question 5: Find the area of a path 20m wide surrounded by a lawn of diameter 200m.

Solution: 

Given:
Width of the path = 20m
Diameter of the circular lawn = 200m
Radius of the inner lawn = diameter/2  
= 200/2
= 100m

The radius of outer circle = radius of inner circle + width of the path
= 100 + 20
= 120m

The annulus formula given by: π (R² – r²)

Where,
R = 120
r = 100
A = π (R - r)(R + r)

A = 3.1415 (120 – 100)(120 + 100)
A = 3.1415 ×20 × 220
A =13822 m²

Thus, the area of the path is 13822 square meter.  

Question 6: Find the radii of the inner and outer circles if the annulus area and the width of the circular field are 2400cm² and 15cm?

Solution: 

Given: A = 2400cm²
Width = 15cm
Width = R – r

Where,
R = radius of outer circle
r = radius of inner circle

Difference in the outer and inner radius = width of the field
R – r = 15
R = 15 + r

The annulus formula given by: A = π (R – r)(R + r)

Substituting the value of R in the annulus formula, we get:
A = π (15 + r – r)(15 + r + r)
A = π (15)(15 + 2r)

The area of the width or the annulus area is 2400cm². Substituting in the above equation, we get:
2400 = π (15)(15 + 2r)
2400 = π (225 + 30r)
225 + 30r = 763.94
30r = 763.94 - 225
30r = 538.94
r = 17.96 cm

The radius of outer circle = r + 15
= 17.96 + 15
= 32.96 cm

Thus, the radii of inner and outer circle are 17.96 cm and 32.96 cm.

Related Articles:

• Area of a Circle
• Geometry of Circles
• Concentric Circles
• Mathematical Formulas for Geometry

Practical Questions

Question 1: Calculate the area of an annulus with an outer radius of 7 cm and an inner radius of 4 cm.

Question 2: Find the area of an annulus if the radii are 10 m and 6 m, respectively.

Question 3: Determine the area of an annulus where the larger circle has a radius of 12 inches and the smaller circle has a radius of 8 inches.

Question 4: An annulus has an area of 25π cm2. If the inner radius is 3 cm, find the outer radius.

Question 5: The radius of the larger circle in an annulus is 15 cm, and the radius of the smaller circle is 10 cm. Calculate the area.

Question 6: Given an annulus with an area of 50π m2 and the outer radius of 9 m, find the inner radius.

Question 7: Calculate the area of an annulus where the outer radius is 20 feet and the inner radius is 14 feet.

Question 8: The area of an annulus is 36π in 2. If the inner radius is 5 inches, find the outer radius.

Question 9: Determine the area of an annulus where the radii of the circles are in the ratio 2:1 and the larger circle's radius is 10 cm.

Question 10: An annulus has an outer radius of 25 meters and an inner radius of 20 meters. Calculate its area.