How to Find the Area of a Semi-Circle Given Its Perimeter

In this article, we will explore the relationship between the perimeter and the area of a semicircle. We will provide detailed steps on how to compute the area of a semicircle using the perimeter, and we will also provide a practical example for clarity.

The Geometry of a Semi-Circle

A semi-circle is essentially half of a circle, which means that its area is half the area of a full circle. The area of a circle is given by the formula:

[ text{Area of a circle} pi r^2 ]

Therefore, the area of a semi-circle is:

[ text{Area of a semi-circle} frac{1}{2} pi r^2 ]

Perimeter of a Semi-Circle

The perimeter of a semi-circle is the sum of the length of its curved part (half the circumference of a full circle) and its diameter. The formula for the perimeter of a semi-circle can be derived as:

[ text{Perimeter of a semi-circle} pi r 2r pi r d ]

Where ( r ) is the radius and ( d ) is the diameter. The radius can be expressed in terms of the diameter as ( r frac{d}{2} ).

Using the Perimeter to Find the Area

Given the perimeter of a semi-circle, we can find the radius and subsequently the area. Here’s a step-by-step guide:

Step 1: Express the Perimeter in Terms of the Radius

The perimeter of a semi-circle is given by:

[ text{Perimeter} pi r 2r r(pi 2) ]

Given the perimeter, we can solve for the radius:

[ r frac{text{Perimeter}}{pi 2} ]

Step 2: Substitute the Radius into the Area Formula

Now that we have the radius, we can substitute it into the area formula:

[ text{Area} frac{1}{2} pi r^2 ]

Rearrange the formula to express the area in terms of the perimeter:

[ text{Area} frac{1}{2} pi left(frac{text{Perimeter}}{pi 2}right)^2 ]

Simplify the equation:

[ text{Area} frac{1}{2} pi left(frac{text{Perimeter}^2}{(pi 2)^2}right) ]

Final simplified form:

[ text{Area} frac{pi text{Perimeter}^2}{2(pi 2)^2} ]

Practical Example

Let’s solve a practical problem where the perimeter of a semi-circle is given. Suppose the perimeter is 20.56 km:

Step 1: Solve for the Radius

Using the perimeter formula:

[ 20.56 pi r 2r r(pi 2) ]

Solve for ( r ):

[ r frac{20.56}{pi 2} approx 4.61 , text{km} ]

Step 2: Calculate the Area

Now, we can find the area using the radius:

[ text{Area} frac{1}{2} pi r^2 frac{1}{2} pi (4.61^2) approx 66.9 , text{km}^2 ]

Conclusion

By understanding the relationship between the perimeter and the area of a semi-circle, we can easily compute the area given the perimeter. The steps involve first finding the radius and then substituting it into the area formula. Whether you are working with a specific radius or the perimeter, the process allows for straightforward area calculation.

Moving forward, you can apply these formulas and steps to solve similar problems involving semi-circles in a variety of contexts. Whether it’s in geometry, calculus, or real-world engineering applications, the principles discussed here remain relevant and useful.