How to find the perimeter of a half circle?
Understand the Problem
The question is asking how to calculate the perimeter of a half circle, which involves understanding the formula for the circumference of a full circle and then adapting it for a half circle while also considering the straight edge of the diameter.
Answer
r(\pi + 2)
Answer for screen readers
The final perimeter of the half circle is $r(\pi + 2)$.
Steps to Solve
- Calculate the circumference of a full circle
Firstly, calculate the circumference of a full circle using the formula:
$$C = 2\pi r$$
where $r$ is the radius of the circle.
- Calculate half of the circumference
Since we need the perimeter of a half circle, halve the circumference calculated in step 1:
$$C_{\text{half}} = \pi r$$
- Add the diameter of the circle
The perimeter of the half circle includes half of the circumference as well as the diameter (which is equal to twice the radius, $d = 2r$). Therefore, add the diameter to the half circumference:
$$P = \pi r + 2r$$
- Simplify the final expression
Combine the terms to get the final perimeter:
$$P = r(\pi + 2)$$
The final perimeter of the half circle is $r(\pi + 2)$.
More Information
The perimeter of a half circle combines both the curved edge and the straight edge (diameter), making it slightly more than half the circumference of a full circle.
Tips
A common mistake is forgetting to add the diameter when calculating the perimeter of the half circle. Always ensure both the curved part and the straight edge are included.