Calculating the Area of a Ring between Concentric Circles

In geometry, a ring, also known as an annulus, is a shape bounded by two concentric circles. This article will explore the calculation of the area of a ring between two concentric circles with radii of 68 cm and 22 cm. We will also address some common misunderstandings and provide a detailed step-by-step solution.

Understanding Concentric Circles

Concentric circles are circles that lie in the same plane and share a common center. In this case, our two circles have a common center and different radii. The radius of the outer circle is 68 cm, while the radius of the inner circle is 22 cm. Our task is to find the area of the region between these two circles.

Area Calculation

The area of the annular ring can be calculated by subtracting the area of the inner circle from the area of the outer circle. Mathematically, this can be expressed as:

Area of the annular ring Area of the outer circle - Area of the inner circle

The area of a circle is given by the formula:

Area πr2

Where r is the radius of the circle.

Step-by-Step Solution

Step 1: Calculate the area of the outer circle.

The radius of the outer circle is 68 cm. Therefore:

Area of the outer circle π(68)2 π × 4624

Step 2: Calculate the area of the inner circle.

The radius of the inner circle is 22 cm. Therefore:

Area of the inner circle π(22)2 π × 484

Step 3: Subtract the area of the inner circle from the area of the outer circle.

Area of the annular ring π × 4624 - π × 484 π × (4624 - 484) π × 4140

Therefore, the area of the annular ring is 4140π square centimeters.

Alternative Calculation Approach

Some sources provide alternative methods for calculating the area of the annular ring. While these methods may yield the same result, it is important to understand the rationale behind them.

Example 1:

Using the same values, we can calculate the area of the annular ring as follows:

Area (68/7)2 - (22/7)2 (682/7) - (222/7) (4624 - 484) / 7 4140 / 7 ≈ 591.43 cm2

Example 2:

Another approach is to directly use the area formula for the annulus (ring):

Area πR2 - πr2 π(R2 - r2)

Where R is the radius of the outer circle and r is the radius of the inner circle. Substituting the values:

Area π(682 - 222) π(4624 - 484) π × 4140 ≈ 13011.43 cm2

Conclusion

In conclusion, the area of the closed figure bounded by the boundaries of the concentric circles with radii of 68 cm and 22 cm is 4140π square centimeters. This method can be used to solve similar problems involving annular regions. Understanding the logic behind these calculations is crucial for accurate and efficient problem-solving in geometry.