What is the Radius of a Semicircle with a Given Circumference?

To find the radius of a semicircle given its circumference, we need to use the formula for the circumference of a semicircle. The formula for the circumference of a semicircle is given by:

Circumference of a Semicircle πr 2r where r is the radius of the semicircle.

Given that the circumference is 54 cm, we can set up the equation as follows:

Setting Up the Equation for Circumference of a Semicircle

Using the formula, we get:

πr 2r 54

Calculating the Radius

We can substitute the value of π as 22/7 for a more precise calculation. This gives:

(22/7)r 2r 54

Multiplying 2 by 7/7 to get a common denominator, we have:

(22r/7) (14r/7) 54

We can then combine the terms:

(36r/7) 54

Solving for r, we get:

r (54 × 7) / 36

Simplifying this, we get:

r (54 × 7) / 36 189 / 36 10.5 cm

Verification and Alternative Methods

Let's verify this by using a few alternative methods to ensure the accuracy of the calculation:

Using π Approximately as 3.14

We can set up the formula again, but using 3.14 as the approximate value of π:

3.14r 2r 54

Simplifying this, we get:

5.14r 54

Solving for r, we get:

r 54 / 5.14 10.5 cm (approximately)

Using π as 22/7 and Simplifying Directly

Recall the formula for the circumference of a semicircle:

(22/7)r 2r 54

This can be solved by combining terms and simplifying:

(22r 14r) / 7 54

Simplifying further:

36r 378

Solving for r:

r 378 / 36 10.5 cm

Conclusion

In conclusion, the radius of a semicircle with a circumference of 54 cm is calculated to be 10.5 cm. This result is consistent across different methods of calculation and confirms the solution.

Additional Information

The semicircle circumference includes the curved part (πr) and the diameter (2r). This total gives us the perimeter of the semicircle. Understanding the formula and how to apply it can be crucial for various real-world applications, such as designing objects or solving geometric problems.