Calculating the Area of a Semicircle Given Its Perimeter

To delve into the problem of finding the area of a semicircle when the total perimeter is given, let's break down the process step by step. This article will provide a comprehensive guide on how to approach such problems, ensuring clarity and proper understanding for students and professionals alike.

Understanding the Perimeter of a Semicircle

The perimeter (P) of a semicircle consists of the curved part (half the circumference of a full circle) and the diameter. The formula for the perimeter of a semicircle is

P πr 2r

where r is the radius of the semicircle.

Setting Up the Equation

Given that the total perimeter is 36 cm, we can set up the equation as follows:

πr 2r 36

Combining and Solving for the Radius

Let's combine the terms by factoring out r:

r(π 2) 36

Now, we solve for r by dividing both sides by (π 2):

r frac{36}{π 2}

Calculate the Area of the Semicircle

The area (A) of a semicircle is given by the formula:

A frac{1}{2} πr^2

Substituting the value of r from the previous step:

A frac{1}{2} π left(frac{36}{π 2}right)^2

Simplifying the expression:

A frac{648π}{π ^2 4π 4}

Conclusion

The area of the semicircle can be calculated using the formula:

A frac{648π}{π ^2 4π 4};text{cm}^2

For a numerical approximation, you can use the value of π ≈ 3.14.

Additional Examples for Understanding

Let's consider a semicircle with a diameter of 10 units. Here, 'units' can be any measure (cm, mm, inches, etc.), but the answer must be in the same squared unit.

Step 1: Calculate the Radius

The radius is half of the diameter:

r frac{10}{2} 5;text{units}

Step 2: Calculate the Area of the Full Circle

Use the formula for the area of a full circle:

A πr^2 3.14 × 5^2 3.14 × 25 78.5;text{units}^2

Step 3: Calculate the Area of the Semicircle

Since the semicircle is half of the full circle, divide the area by 2:

A frac{78.5}{2} 39.25;text{units}^2

Units and Reporting

Remember to include the units squared in your final answer. If the diameter was given in a unit (like KaT, cm, in, etc.), the area should be reported as units squared.

Key Points to Remember

Always solve for r first before finding the area. Make sure to include units squared in your final answer. For numerical approximations, use π ≈ 3.14 if two decimal places are sufficient.

By following these steps, you can calculate the area of a semicircle with ease and precision.