Annulus Introduction
An Annulus is a ring-shaped object that is defined by the circumference of two concentric circles of different radii. It has a ring-like flat form. Finding the areas of the outer and inner circles can be used to compute the annulus' area. To achieve the answer, we must find the difference between the areas of both circles. The ring's outer and inner circles are concentric, and they share a single centre point. In this article, we will learn to calculate the area of the ring formula with the help of the area of ring and practise a few solved examples.
Annulus Meaning
The word "annulus" (plural – annuli) comes from the Latin word "annulus," which literally means "small ring." The area between two concentric circles is known as an annulus.
It's the area enclosed by two circles that share the same centre. This form is similar to a flat ring. It's also known as a circular disc with a round hole in the centre. An annulus is depicted in this diagram.
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Two circles may be seen here, with a smaller circle inside the larger one. In the given diagram, we can see the centre of both circles is the point O. An annulus is the shaded coloured area between the boundaries of these two circles. The inner circle is the smaller of the two circles, while the outer circle is the larger.
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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. Subtracting the area of a smaller circle from the area of a larger circle gives the circular ring formula.
Given below is annulus formula :
\[ A = (R^{2} - r^{2})\]
\[R = \sqrt{r^{2} + \frac{A}{\Pi}}\]
\[r = \sqrt{R^{2} - \frac{A}{\Pi}}\]
Where,
A = Area of Annulus
R = Outer radius
r = Inner radius
(Pi) π = is approximately 3.142
Area of a Circular Ring
The surface area of a circular ring is bounded by two concentric circles with radius R and r (R > r).
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We know the area of concentric circle is the area between the two given concentric circles that can be calculated by subtracting the area of the inner circle from the area of the outer circle.
Hence, area of ring formula \[ = \Pi \times R^{2} - \Pi \times r^{2}\]
\[ = \Pi (R^{2} - r^{2})\]
\[ = \Pi (R + r) (R - r)\]
Surface area of a ring is the same as the area of circular ring .
Perimeter of Ring
The annulus is a flat circular shape made up of two concentric circles that can also be thought of as a ring. As a result, an open ring can be compared to a cylinder and a punctured plane from a topological standpoint. To get the perimeter of the annulus, we must examine both the inner circle and the outer circle, just as we did with the area. As a result, the ring's or annulus' perimeter is equal to the sum of the radii of the large and small circles multiplied by two.
Perimeter of ring (P) = \[ 2 \times \Pi \times (R + r)\]
Here the outside radius is R, and the inner radius is r.
Solved Examples:
1. A circular path's outer and inner diameters are 728 and 700 metres, respectively. Calculate the circumference and area of the circular path.
Sol: The outer radius of a circular path \[ R = \frac{728}{2}\] = 364m.
The inner radius of a circular path \[ r = \frac{700}{2}\] 350m.
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Therefore, breadth of the circular path will be R - r = 364 m - 350 m = 14 m.
We know the area of the circular path is \[ \pi (R + r)(R - r) \]
\[\frac{22}{7} \times (364 + 350) (364 - 350)m^{2}\]
\[\frac{22}{7} \times 714 \times 14 m^{2}\]
\[ 22 \times 714 \times 2m^{2}\]
\[31416 m^{2}\]
Hence, the area of the circular path is \[31416 m^{2}\].
2. Find the radii of the inner and outer circles if the area of an annulus is 1092 inches and the width is 3 cm.
Sol: An annulus has an inner radius of r and an outer radius of R.
Then width = R – r
3 = R – r
R = 3 + r
We know,
Area of the annulus =\[\Pi (R^{2} - r^{2})\]
Or we can also write as \[ \pi (R + r) (R – r) \]
Now, put value in above equation
1092 = \[ \frac{22}{7} \times (3 + r + r) \times 3 \]
\[ \Rightarrow 3 + 2r = 1092 \times 722 \times 3\]
\[ \Rightarrow 3 + 2r = 115.82\]
\[ \Rightarrow 2r = 115.82 - 3 \]
\[ \Rightarrow 2r = 112.82 \]
\[ \Rightarrow r = \frac{113.82}{2} \]
\[ \Rightarrow r = 56.41\]
So value of R = 3 + 56.41
= 59.41
Hence, Inner radius = 56.41 inches
Outer radius = 59.41 inches
FAQs on Annulus Formula
1. Define Concentric Circle.
Concentric circles are circles that share a common center point. By inverting any two circles and selecting the inversion center as one of the limiting points, any two circles can be made concentric.
2. What is the area of Annulus?
Finding the areas of the outer and inner circles can be used to compute the annulus' area. Then, to get the answer, we must subtract the areas of both circles.
3. What is the formula for calculating the annulus area?
The formula used for finding the area of the annulus is \[\Pi (R^{2} - r^{2})\] square units, or it can be written as Area of Annulus = \[ \pi(R+r)(R-r)\] square units. Here R is the radius of the outer circle, r is the radius of the inner circle.
