To find the perimeter of a square that circumscribes a circle of radius $a$, let's go through the steps.

Step 1: Understand the relationship between the square and the circle

In this case, the square is circumscribed around the circle. This means that the circle is inscribed within the square, touching the square at exactly four points (the midpoints of the square's sides). The circle's diameter is equal to the length of the square's side.

Step 2: Relationship between radius and side length

The radius of the circle is $a$. The diameter of the circle is twice the radius, so the diameter is: $\text{Diameter} = 2a$ Since the diameter of the circle is equal to the side length of the square (because the circle fits perfectly within the square), the side length of the square is also $2a$.

Step 3: Calculate the perimeter of the square

The perimeter $P$ of a square is given by the formula: $P = 4 \times \text{side length}$ Substitute the side length of the square: $P = 4 \times (2a) = 8a$

Final Answer:

The perimeter of the square circumscribing a circle with radius $a$ is $\boxed{8a}$.

Would you like further explanation on any part of the process or additional examples? Here are some related questions:

1. How does the perimeter of the square change if the radius of the circle is doubled?
2. How would you find the area of the square circumscribing a circle?
3. What would be the perimeter of a square inscribed in a circle of radius $a$?
4. How do the side lengths of squares compare in different geometric configurations involving circles?
5. What is the relationship between the area of a circle and the side length of a square circumscribed around it?

Tip: Always be mindful of whether the circle is inscribed in or circumscribed around the square, as this affects the relationships between their dimensions!