How to Find the Area of an Annulus with Formula and Examples
Think about the last time you had a glazed doughnut. Yummy! If your teacher asked you the shape of your glazed doughnut, what would you say? You may be thinking, Well when I look at it from above, it is in the shape of a circle! But let's think about this. Circles do not have holes in the middle of them. So what type of shape would your doughnut be? What is the shape that looks like a circle but has a hole in the middle of it? The shape is called an annulus! Let us discuss the annulus meaning?!
An annulus or the annular region can basically be defined as a shape that is made out of two circles.
It is a plane figure that is formed by two concentric circles.
The region covered between two concentric circles is known to be the annulus or the annular region.
The annulus has a ring shape and it has many applications in Mathematics. Some of the real-life examples of an annulus are dough-nut,finger-ring etc.
The area of the annulus or the annular region can be determined if we know the area of both the circles (both inner and outer) that form the annulus.
Formula of Annulus
The formula to find annulus of the circle can be given by:
where ‘R’ is known to be the radius of outer circle
‘r’ is known to be the radius of the inner circle of the annulus.
In this article we are going to know what annulus is, the area and examples.
The circle is known to be a fundamental concept not only in Mathematics but it is also considered as an important concept in many fields. By its definition, we know that a circle is a plane figure which is generally made up of the points that are situated at the same distance from a particular point. See the figure of a circle given below:
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The picture above shows a complete circle with some radius denoted by r. Now if the same circle that we see above is surrounded by another circle with some space in between the two and the radius of the new circle is bigger than this circle, then the region formed in between the two circles is basically known as the annulus or the annular region. Let us learn the meaning of annulus in terms of geometry along with the area formula and solved examples based on the topic.
What Does The Term Annulus Mean?
The word “annulus” is derived from the Latin word.
The word annulus or annular meaning is “little ring“.
An annulus is known to be the area between two circles that are concentric ( that is circles whose centre coincide) lying in the same plane.
The plural of the term annulus is – annuli.
It can be defined as a region bounded between two circles that are concentric (that is they share the same centre). The shape of the annulus resembles a flat ring. Annulus can also be considered as a circular disk that has a circular hole in the middle. See the figure here showing an annulus given below.
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Here,we can see two circles ,where a small circle lies inside the bigger one both having different radius. The point O is known to be the centre of both circles. The shaded coloured area, between the boundary of these two circles (the bigger one and the smaller one), is known as an annulus. The smaller circle is known to be the inner circle meaning the smaller one, while the bigger circle is termed as the outer circle.
In simpler words, any two-dimensional flat ring-shaped object which is formed by two concentric circles is known as an annulus.
What is The Area of Annulus?
We can find the area of the annulus by finding the area of the outer circle and the inner circle meaning the smaller one. Then we need to subtract the areas of both the circles to obtain the result. Let us consider the figure given below:
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In the figure above, the two circles have common centre O. We will let the radius of outer circle be equal to “R” and the radius of inner circle meaning the smaller one be equal to “r”. The shaded portion indicates a space known as the annulus. To find the area of this annulus, we are required to find the areas of the two circles.
Therefore,
Area of Outer big Circle = π\[R^{2}\]
Area of Inner small Circle = π\[r^{2}\]
Therefore, Area of Annulus = Area of Outer big Circle – Area of Inner small Circle
Hence,
Or this can also be written as;
Questions to be solved
Question 1) If the area of an annulus is equal to 1092 inches and its width is equal to 3 cm, then find the radii of the inner circle and outer circle.
Solution) Let the inner radius of an annulus be equal to r and its outer radius be equal to R.
Then width will be equal to R – r
R – r = 3
R = 3 + r
We know that the formula for calculating the area of annulus,
Area of Annulus is equal to π(\[R^{2}\] - \[r^{2}\])
or
Area of the annulus is equal to π (R + r) (R – r)
22/7 (3 + r + r) (3) = 1092
1092×722×3 = 3 + 2r
115.82 = 3 + 2r
115.82 – 3 = 2r
2r = 112.82
Therefore, the value of r = 56.41
Then the value of R = 3 + 56.41 = 59.41
So, the value of the Inner radius = 56.41 inches
And the outer radius = 59.41 inches
FAQs on Annulus in Geometry Definition Area and Properties
1. What is an annulus in maths?
An annulus is the region between two concentric circles with different radii. It is a ring-shaped area formed when a smaller circle is removed from a larger circle that shares the same center. In geometry, an annulus is defined by:
- An outer radius (R)
- An inner radius (r), where r < R
2. What is the formula for the area of an annulus?
The area of an annulus is given by the formula A = π(R² − r²). Here:
- R = outer radius
- r = inner radius
- π ≈ 3.1416
3. How do you calculate the area of an annulus step by step?
To calculate the area of an annulus, subtract the area of the inner circle from the outer circle. Follow these steps:
- Step 1: Identify R and r
- Step 2: Square both radii (R² and r²)
- Step 3: Subtract: R² − r²
- Step 4: Multiply by π
- A = π(25 − 9)
- A = π × 16
- A = 16π cm²
4. What is the difference between a circle and an annulus?
A circle is a single round region enclosed by one boundary, while an annulus is the region between two concentric circles. Key differences:
- A circle has one radius and one boundary.
- An annulus has two radii: inner radius (r) and outer radius (R).
- The area of a circle is πr², while the area of an annulus is π(R² − r²).
5. What is the perimeter of an annulus?
The perimeter of an annulus is the sum of the circumferences of the outer and inner circles. The formula is:
- Outer circumference = 2πR
- Inner circumference = 2πr
- Total perimeter = 2πR + 2πr
6. Can you give an example of an annulus in real life?
A common real-life example of an annulus is a ring or a washer. Other examples include:
- A doughnut shape
- A CD or DVD
- A circular running track lane
7. How do you find the inner radius of an annulus if the area is given?
To find the inner radius (r), rearrange the annulus area formula A = π(R² − r²). Steps:
- Step 1: Divide both sides by π → A/π = R² − r²
- Step 2: Rearrange → r² = R² − A/π
- Step 3: Take the square root → r = √(R² − A/π)
8. What are the properties of an annulus?
The main properties of an annulus describe its geometric structure and measurements. Important properties include:
- It is formed by two concentric circles.
- It has two radii: inner (r) and outer (R).
- Its area is π(R² − r²).
- Its total boundary length is 2πR + 2πr.
- It is symmetrical about its center.
9. Why is the area of an annulus π(R² − r²)?
The area of an annulus is π(R² − r²) because it equals the area of the larger circle minus the smaller circle. Mathematically:
- Area of outer circle = πR²
- Area of inner circle = πr²
- Annulus area = πR² − πr²
- Factor out π → π(R² − r²)
10. What is the difference between an annulus and a circular ring sector?
An annulus is the full ring between two concentric circles, while a circular ring sector is only a portion of that ring. Differences include:
- An annulus covers 360° around the center.
- A ring sector covers only a specific central angle (θ).
- Area of annulus = π(R² − r²).
- Area of ring sector = (θ/2)(R² − r²) when θ is in radians.
