Annulus Formula - TRUNG TÂM GIA SƯ TÂM TÀI ĐỨC

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An Annulus is a ring-shaped object, bounded by the circumference of two concentric circles of two different radii. An Annulus is much like the throw-ring. One way to think of it is a circular disk with a circular hole in it. The outer and inner circles that define the ring are concentric, that shares a common center point.

The dimensions of an annulus are defined by the two radii R, r, which are the radii of the outer ring and the inner ‘hole’ respectively. The area of a circular ring can be found by subtraction the area of a small circle from that of the large circle.

Here are formulas to find Area of Annulus.

Where,
A = Area of Annulus
R = Outer radius
r = Inner radius

Annulus

An annulus is an inner region between two concentric circles i.e. two or more circles sharing the same center point. The annulus is shaped like a ring and has many applications in mathematics that we will be learning in this article. Some of the real-life examples are a doughnut, finger rings. etc. Let us learn more about the shape of the annulus and solve a few examples to understand the concept better.

Annulus Definition

An annulus is a two-dimensional flat figure shaped in a circular form which is constructed by two concentric circles. The region or the area formed in between these two concentric circles is called the annulus. Since it is a flat figure in a circular form, the edges are two circles with the same center. It is considered a circular disk having a circular hole in the middle.

Annulus Meaning

The word annulus is derived from a Latin word, ‘annuli’, meaning little rings. The shape of the annulus is flat and circular with a hole in between, much like a throw ring or a circular disc. Look at the image below showing two circles i.e. one small circle also called an inner circle and a big circle also called the outer circle. The point which is marked as red is the center of both circles. The shaded colored area, between the boundary of these two circles, is known as an annulus.

Area of the Annulus

The annulus area is the area of the ring-shaped space i.e. the enclosed region between the two concentric circles. To calculate the area of the annulus, we need the area of both the inner circle and the outer circle. The dimensions of an annulus are defined by the two radii R, and r, which are the radii of the outer ring and the inner ring respectively. Once the measurements of both the radii are known, we can calculate the area by subtracting the area of the small circle from the big circle. Hence, the formula used for finding the area of the annulus is:

Area of Outer Circle = πR²

Area of Inner Circle = πr²

Area of Annulus = Area of Outer Circle – Area of Inner Circle

Therefore, Area of Annulus = π(R²-r²) square units, or it can be written as π(R+r)(R-r) square units, where R is the radius of the outer circle, r is the radius of the inner circle, and π(pi) is approximately 3.142. Look at the image below, the area of the outer (bigger) circle – the area of the inner (smaller) circle = the area of the annulus.

Annulus Perimeter

The perimeter is the distance around the 2D shape. Since the annulus is a flat circular shape constructed by two concentric circles, it can also be considered as a ring. Therefore, an open ring can be considered as the topological equivalent of a cylinder and a punctured plane. Similar to the area, to find the perimeter of the annulus we need to consider both the inner circle and the outer circle. So, the perimeter of the ring or annulus is equal to the sum of the radii of the large and small circles multiplied by 2π. The formula for finding the perimeter is:

Perimeter of Annulus (P) = 2π(R+r) units, where R is the radius of the outer circle, r is the radius of the inner circle, and π (pi) is approximately 3.142.

Annulus Examples

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

Solution: Given that outer radius (R) = 15 units and inner radius (r) = 8 units.Area of outer circle = πR² = 3.142 × 15 × 15 = 706.95 unitsArea of inner circle = πr² = 3.142 × 8 × 8 = 201.088 unitsThe formula to find the area of an annulus is (area of the outer circle – area of the inner circle) square units.Area = (706.95 – 201.088) square unitsTherefore, the required area is 505.862 square units.

Example 2: If the outer radius of a ring is 10 units and the inner radius of a ring is 3 units, what would be the perimeter of the ring?

Solution: We already know that the perimeter of the annulus = 2π(R+r) and it is given that R = 10 units and r = 3 units.P = 2 × 3.142 (10 + 3)P = 6.284 × 13P = 81.692 unitsTherefore, the perimeter of the ring is 81.692 units.

Example 3: A steel pipe has an outer diameter of 80 units and an inner diameter of 60 units, what is the area of the cross-section?

Solution: As we know the diameter is twice the radius. So we divide the value by 2 to obtain the radius.Outer radius, R = 80/2 = 40 unitsInner radius, r = 60/2 = 30 unitsArea of Annulus = π(R²-r²) square unitsA = 3.142 (40² – 30²)A = 3.142 (1600 – 900)A = 3.142 × 700A = 2199.4 units²Therefore, the area of the cross-section of the pipe is 2199.4 units².

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)

= 848230 m²

Thus, the area of the circular track is 848230 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)

= 306296 m²

Thus, the area of the circular track is 306296 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 circle 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 circle 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 = 764
30r = 764 – 225
30r = 539
r = 18cm (approx)
The radius of outer circle = r + 15
= 18 + 15
= 33cm
Thus, the radii of inner and outer circle are 18cm and 33cm.
Q.1: Calculate the area of an annulus whose outer radius is 14 cm and inner radius 7 cm?
Solution: Given that outer radius R = 14 cm and inner radius r = 7 cm
Area of outer circle = πR² = 22/7 x 14 x 14
= 22 x 14 x 2
= 616 cm²
Area of inner circle = πr² = 22/7 x 7 x 7
= 22 x 7
= 154 cm²
Area of the annulus = Area of the outer circle – Area of the inner circle
Area of the annulus = 616 – 154
Area of the annulus = 462 cm²
Q.2: If the area of an annulus is 1092 inches and its width is 3 cm, then find the radii of the inner and outer circles.
Solution: Let the inner radius of an annulus be r and its outer radius be R.
Then width = R – r
3 = R – r
R = 3 + r
We know,
Area of the annulus = π(R²−r²)
or
Area of the annulus = π (R + r) (R – r)
1092 = 22/7 (3 + r + r) (3)
3 + 2r = 1092×722×3
3 + 2r = 115.82
2r = 115.82 – 3
2r = 112.82
r = 56.41
R = 3 + 56.41
= 59.41
So, Inner radius = 56.41 inches
Outer radius = 59.41 inches

FAQs on Annulus

What is the Meaning of the Word Annulus?

The word annulus is derived from the Latin word annuli which means little rings. It is a 2D flat figure which is circular in nature but a hole in between like a doughnut. An annulus is the region enclosed between the boundary of these two circles.

What is Annulus Area?

The annulus area is the area of the ring-shaped i.e. the region enclosed between the two concentric circles. To calculate the area of the annulus, the areas of both the inner circle and the outer circle are required.

Area of Outer Circle = πR²

Area of Inner Circle = πr²

Based on these, the formula for the area of the annulus is area of outer circle – area of inner circle. Therefore, Area of Annulus = π(R²-r²) square units, where R is the radius of the outer circle, r is the radius of the inner circle, and π (pi) is approximately 3.142.

What is the Perimeter of the Annulus?

Annulus perimeter is considered as the entire distance around the 2D shape including both the inner circle and the outer circle. The formula to find the perimeter of the annulus is 2π(R+r) units, where R is the radius of the outer circle, r is the radius of the inner circle, and π (pi) is approximately 3.142.

Is a Circle an Annulus?

In mathematics, an annulus is a region between two concentric circles. It is shaped like a ring or a donut. A circle is not an annulus since it is a completely closed figure in a circular manner, whereas an annulus is a circular-shaped figure but with a hole in between.

What is Annulus Radius?

The annulus radius is the difference between the radius of the outer circle and the inner circle. Since an annulus is constructed by two concentric circles, the radius of both the circles is required. The dimensions of the annulus radius are derived from the radii of the outer circle and inner circle with R and r as the measures respectively.

What Dimension is an Annulus?

An annulus is a two-dimensional shape that is flat and circular in nature. It is constructed by two concentric circles with a hole in between and that enclosed region is called the annulus.