Calculating Semicircle Diameter from Perimeter

The question asks us to find the diameter of a semicircle given its perimeter and the value of $\pi$. We are given:

• Perimeter of the semicircle = $25.7$ cm
• Value of $\pi$ = $3.14$

Let the diameter of the semicircle be $d$ cm. The perimeter of a semicircle is the sum of the length of the curved arc (which is half the circumference of a full circle) and the length of the straight edge (which is the diameter).

The circumference of a full circle with diameter $d$ is given by the formula $C = \pi d$.

The length of the curved arc of the semicircle is half of the circumference: $\frac{1}{2} \times \pi d$.

The straight edge is the diameter, which is $d$.

Therefore, the perimeter of the semicircle is given by the formula:

$\text{Perimeter} = (\text{Curved Arc}) + (\text{Straight Edge})$

$\text{Perimeter} = \frac{1}{2}\pi d + d$

We can factor out $d$ from the formula:

$\text{Perimeter} = d \left(\frac{\pi}{2} + 1\right)$

Now, we substitute the given values into the formula:

$25.7 = d \left(\frac{3.14}{2} + 1\right)$

First, calculate the value inside the parenthesis:

$\frac{3.14}{2} = 1.57$

So, the equation becomes:

$25.7 = d (1.57 + 1)$

$25.7 = d (2.57)$

To find the diameter $d$, we need to divide the perimeter by $2.57$:

$d = \frac{25.7}{2.57}$

Performing the division:

$d = 10$

The diameter of the semicircle is $10$ cm.

Now let's check the given options:

• Option 1: $8$ cm
• Option 2: $12$ cm
• Option 3: $10$ cm
• Option 4: $9$ cm

Our calculated diameter of $10$ cm matches Option 3.

Revision Table for Semicircle Perimeter

Additional Information on Semicircle Calculations

A semicircle is essentially half of a circle. When dealing with semicircle problems, it's crucial to distinguish between its area and its perimeter. The area is straightforward - it's half the area of the corresponding full circle.

However, the perimeter is often where mistakes happen. The perimeter includes the curved part (the arc) and the straight part (the diameter). Think of it as walking around the edge of the shape. You walk along the curve and then back across the straight line that cuts the circle in half.

The value of $\pi$ is approximately $3.14159$. In many problems, a rounded value like $3.14$ or $\frac{22}{7}$ is provided to simplify calculations. Always use the value given in the question.

Understanding the relationship between radius ($r$) and diameter ($d$), where $d=2r$, is also fundamental for these types of geometry problems.