How to Find the Area of an Annulus with 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 What Is an Annulus in Maths?
1. What is an annulus in mathematics?
An annulus is a flat, ring-shaped geometric figure. It is defined as the region on a plane that lies between two concentric circles—circles that share the same centre point but have different radii. Visually, an annulus looks like a circular disk with a smaller, concentric circular hole in its centre.
2. How is the area of an annulus calculated?
The area of an annulus is calculated by subtracting the area of the inner circle from the area of the outer circle. The formula is:
Area = π (R² - r²)
Where:
- R is the radius of the outer circle.
- r is the radius of the inner circle.
- π (pi) is a mathematical constant, approximately equal to 3.14159.
3. What are the key properties that define an annulus?
The primary properties that define an annulus are:
- It is a two-dimensional plane figure.
- It is formed by two concentric circles, which means they must share the exact same centre point.
- It has an outer radius (R) and an inner radius (r), where the outer radius is always greater than the inner radius (R > r).
- The width of the ring is constant and is equal to the difference between the two radii (R - r).
4. What are some real-world examples of an annulus shape?
The annulus shape appears in many common objects. Key examples include:
- A metal washer or a rubber gasket used in hardware.
- The flat, data-storing surface of a CD, DVD, or Blu-ray disc.
- The cross-section of a hollow pipe or tube.
- A circular running track around a sports field.
- The visual effect of an annular solar eclipse, where the Moon blocks the Sun's centre, leaving a bright ring of fire.
5. Can the two circles that form an annulus have different centres?
No, by definition, the two circles that form an annulus must be concentric, meaning they must share the exact same centre point. If the two circles have different centres, the resulting shape is not a true annulus but an eccentric ring, where the thickness of the ring is not uniform.
6. How is an annulus fundamentally different from a regular circle or a disk?
The key difference lies in what each term describes. A circle is a one-dimensional curved line (the circumference) with no area. A disk is the two-dimensional area enclosed by a single circle. In contrast, an annulus is the two-dimensional area bounded by two distinct concentric circles, creating a ring shape with a hollow centre.
7. Where does the term "annulus" come from?
The term "annulus" originates from Latin. It is derived from the Latin word anulus, which translates to "little ring". This name is a direct and accurate description of the geometric shape, making its etymology easy to remember.
8. For which grade level, such as Class 10, is understanding the annulus important according to the CBSE syllabus?
The concept of an annulus is most relevant for CBSE Class 10 Maths, within the chapter on "Areas Related to Circles". As per the 2025-26 syllabus, students are expected to solve problems involving areas of combinations of plane figures. Calculating the area of a circular path or track is a classic application problem that directly uses the annulus formula.
9. What common mistake should students avoid when calculating the area of an annulus?
A frequent mistake is to subtract the radii first and then square the result, i.e., attempting to use the formula π(R - r)². This is incorrect and will lead to the wrong answer. The correct procedure is to square each radius individually first and then subtract the smaller squared value from the larger one. Always use the correct formula: Area = π(R² - r²).
