 In this article, you will learn about how to find the Area of a Circular Ring. A circular ring is a plane figure which is bounded by the circumference of two concentric circles of two different radii. A circle is made of multiple points arranged equidistant from a single central point and that point is called the center of the circle. The outer circle and inner circles define that the ring is concentric, that shares a common center point. The best way to think of it is a circular disk with a circular hole in it.

On this page, we will discuss the definition of the area of a circular ring, formulas, solved example problems, and so on.

Area of a Circular Ring – Definition

The area of a circular ring will be subtracted from the area of a large circle to the area of a small circle. A ring-shaped object is bounded by the circumference of two concentric circles of two different radii. The dimensions of the two radii are R, r, which are the radii of the outer ring and the inner ring respectively. Formula:

To find the area of a circular ring or annulus, to multiply the product of the sum and the difference of the two radii. It will be bounded by two concentric circles of radii R and r (R>r).
Therefore, the area of a circular ring is, area of the bigger circle – the area of the smaller circle.
= π(R + r) (R – r) = πR²- πr²
where R and r are the outer circle radius and the inner circle radius. So, it will be as R=√ r2+Aπ  and r = √ R2+Aπ.

Area of a Circular Ring Examples

Problem 1:

Find the area of a flat circular ring formed by two concentric circles, circles with the same center whose radii are 8cm and 4cm? Solution:
Given in the question, the values are
The radius (r1) of the bigger circle is 8cm.
The smaller circle radius (r2) is 4cm.
Now, we have to find the area of a circular ring.
As we see the required area is between the two circles as shown in the figure.
Using the formula, we can find the value.
The area of the shaded portion is = Area of the bigger circle – Area of the smaller circle.
A = πR²- πr²
Substitute the given values within the above formula, we get
A = π(8×8) – π(4×4) = 64π – 16π = 48π
48π = 48 x 22/7 = 6.85 x 22 = 150.7 cm²
Therefore, the area of the circular ring formed by two concentric circles is 150.7 cm².

Problem 2:

A path is 21cm wide surrounds a circular lawn with a diameter of 240cm. Find the area of the path?

Solution:
As given in the question,
A circular dawn diameter is 240cm.
So, the radius of the inner circle(r) is 120 cm.
The wide of a path is 21cm.
The radius of the outer circle(R) is 120+21 = 141cm.
Now, we have to find the area of a path.
We know the formula, Area of a path is π(R+ r) (R – r).
After the substitution of the value, we get,
Area of the path = 22/7(141+120)(141-120).
= 22/7(161)(21) = 22 x 23x 21=10616Sq.cm
Thus, the Area of the path is 10616sq.cm.

Problem 3:

The inner diameter and the outer diameter of the circular path are 628 m and 600m respectively. Find the breadth of a circular path and the area of the circular path. Consider the π value as 22/7. Solution:
As given in the question,
The outer radius of a circular path (R) is, 628/2 = 314m.
The inner radius of a circular path (r) is, 600/2 = 300m.
Now, we have to find the breadth of a circular path and the area of a circular path.
So, the breadth of a circular path is R-r = 314m – 300m = 14m.
Next, the Area of a circular path is, π(R + r)(R – r)
Area = 22/7 (314+300)(314-300)
A= 22/7(614)(14)m² = 27016m².
Therefore, the area of the circular path is 27016m².

Problem 4:

Find the area of a circular ring formed by two concentric circles whose radii are 6.2cm and 5.8cm respectively. Take the π value as 3.14.

Solution:
As we know the radii of the outer circle and inner circles will be R and r respectively.
The concentric circle (R) value is 6.2 cm and the concentric circle (r) value is 5.8 cm.
Now, we will find the area of a circular ring value.
We know the formula, Area of a circular ring is π(R+ r) (R – r).
Substitute the given values in the above formula, we get
A=π(R+ r) (R – r) = 22/7 (6.2 +5.8)(6.2 – 5.8)
sq. cm
sq. cm
Hence, the area of a circular ring value is 15.072 sq. cm

Problem 5:
A circular ring is 8cm wide. Find the difference between the outer circle radius and inner circle radius?

Solution:
As given in the question, the circular ring wide is 8cm.
We have to find the difference value of the outer circle and inner circle radius. The difference between the outer circle radius and inner circle radius is R-r.
So, the value of R-r is 8 cm.
Thus, the difference value of an inner circle radius and an outer circle radius is 8cm.