Annulus

Introduction

Annulus (plural annuli or annuluses)The area between two concentric circles. The Latin word annulus, or annulus, which means "small ring," is where the name "annulus" originates. Annular is the adjectival form (as in annular eclipse). The open annulus and the perforated plane are topologically equal to one another. An annulus is an interior space between two concentric circles or circles with more than two centres of rotation. This article will teach you about the several mathematical uses of the annulus, which has a ring-like shape. Examples from everyday life include finger rings, doughnuts, and more. For a better understanding of the idea, let's study more about the annulus's shape and work through a few examples.

Annulus: Circular Ring (Definition)

• A two-dimensional flat figure with a circular shape called an annulus is made up of two concentric circles.

• The annulus is the region or area created between these two concentric rings. The edges are two circles with the same centre since it is a flat figure with a circular shape.

• It is seen as a circular disc since the centre of the object has a circle.

• The Latin word "annuli," which means "small rings," is where the word "annulus" originates.

• The annulus resembles a throw ring or a circular disc since it is flat and circular with a hole in the middle.

• Look at the illustration below, which features two circles—one little, also known as the inner circle, and one large, generally known as the outer circle. Both circles' centres are located at the black-marked location. An annulus is a region between these two circles' edges that is green in colour.

Perimeter & Area

• The 2D shape's per

• imeter is the space surrounding it. The annulus can also be referred to as a ring because it is a flat circular form made up of two concentric circles. As a result, a punctured plane and a cylinder can be thought of as the topological equivalent of an open ring. Similar to the area, we must take into account both the inner and outer circles in order to get the annulus' perimeter. The circumference of the ring or annulus is then determined by adding the radii of the large and small circles and multiplying the result by 2. To calculate the perimeter, use the following formula −

  $$\mathrm{Perimeter\: of\: annulus=2π(R+r)}$$

  where R is the radius of the outer ring and r is the radius of the smaller ring.

• The area of the ring-shaped space, or the enclosed area between the two concentric circles, is known as the annulus area. We must know the areas of the inner and outer circles in order to determine the annulus's area. The two radii R and r, which correspond to the radii of the outer and inner rings, respectively, define the dimensions of an annulus. When we know the dimensions of both radii, we can compute the area by deducting the area of the tiny circle from the area of the large circle. Consequently, the following formula is used to determine the annulus's area −

  $$\mathrm{Area\: of\: annulus=π(R^2-r^2)}$$

  where R is the radius of outer ring and r is the radius of the smaller ring.

Solved Examples

1)If the annulus' outer radius is 12 units and its inner radius is 6 units, calculate its area.

Answer:

Given that

$$\mathrm{R=12\: \&\: r=6}$$

Thus employing the formula of area of an annulus we get;

$$\mathrm{Area\: of\: annulus=π(R^2-r^2 )}$$

$$\mathrm{=π(12^2-6^2 )}$$

$$\mathrm{=π(144-36)=108π\: sq. units}$$

2)Calculate the perimeter of the annulus if the outer radius is 14 units and the inner radius is 9 units.

Answer:

Given that

$$\mathrm{R=14\: \&\: r=9}$$

Thus employing the formula of the Perimeter of an annulus we get;

$$\mathrm{Perimeter\: of\: annulus=2π(R+r)=2π(14+9)=2π(23)=46π\: units}$$

3)What is the area of the cross-section of a steel pipe with an outer radius of 100 units and an inner radius of 50 units?

Answer:

Given that

$$\mathrm{R=100\: \&\: r=50}$$

Thus employing the formula of area of an annulus we get;

$$\mathrm{Area\: of\: annulus=π(R^2-r^2)=π(100^2-50^2)=π(150)(50)=7500π\: sq. units}$$

Thus, the area of the cross-section of a steel pipe is 7500π sq. units

4)Find the inner radius of the annulus whose perimeter is 18π and outer radius is 7 cm

Answer:

Given that

$$\mathrm{R=7\: \&\: r=?}$$

Thus employing the formula of Perimeter of an annulus we get;

$$\mathrm{Perimeter\: of\: annulus=2π(R+r)=2π(7+r)}$$

$$\mathrm{18π=2π(7+r)}$$

$$\mathrm{\frac{18}{2}=r+7}$$

$$\mathrm{r=9-7=2}$$

Thus, the radius of the inner circle is 2 cm.

5)What is the area of the DVD disc with an outer radius of 100 units and an inner radius of 50 units?

Answer:

Given that

$$\mathrm{R=100\: \&\: r=50}$$

Thus employing the formula of area of an annulus we get;

area of annulus=π(R²-r²)=π(100²-50²)=π(150)(50)=7500π sq. units. Thus the area of the DVD disc is 7500π

6)If the annulus' outer radius is 14 units and its inner radius is 7 units, calculate its area.

Answer:

Given that

$$\mathrm{R=14\: \&\: r=7}$$

Thus employing the formula of area of an annulus we get;

Area of annulus=π(R²-r²)=π(14²-7²)=π(196-49)=147π sq. units

7)Find the inner radius of the annulus whose perimeter is 20π and outer radius is 8 cm

Answer:

Given that

$$\mathrm{R=8\: \&\: r=?}$$

Thus employing the formula of Perimeter of an annulus we get;

$$\mathrm{Perimeter\: of\: annulus=2π(R+r)=2π(8+r)}$$

$$\mathrm{20π=2π(8+r)}$$

$$\mathrm{\frac{20}{2}=r+8}$$

$$\mathrm{r=10-8=2}$$

Thus, the radius of the inner circle is 2 cm.

Conclusion

• An annulus is a two-dimensional flat figure with a circular shape formed by two concentric circles. The annulus is the region or area formed between these two concentric circles.

• Area of annulus=π(R²-r²) where R is the radius of the outer ring and r is the radius of the smaller ring.

• Perimeter of annulus=2π(R+r) where R is the radius of the outer ring and r is the radius of the smaller ring.

FAQs

1. An annulus has how many sides?

An annulus is a flat shape that resembles a ring. It has two circles with the same centre as its edges. An annular object is one that has the shape of an annulus.

2. What is the definition of annular?

An annulus is a shape formed by joining two circles. An annulus is a plane figure formed by the intersection of two concentric circles (the circles sharing a common center). The annulus has the shape of a ring.

3. What part of the circle is the annulus?

The annulus is the space between the two circles' outer edges. The area between the two circles' outer edges will always resemble a circle with a hole in the centre. The difference of two concentric circles forms an annulus.

4. Is it possible for an annulus to be square?

Depending on how it is sliced, the annulus can be interpreted as the square of a closed-string one-point function or an open-string two-point function.

5. What exactly is an annular surface?

An annular surface is a surface with a single system of curvature lines made up of circles. Consider the spheres that touch two consecutive surface spheres: these spheres form a special linear congruence; they must touch the sphere.