Question

What would be the circumference of a circle that has been inscribed in a square of area 5.

• A. 3\(\pi\)
• B. 5\(\pi\)
• C. \(\sqrt{5} \pi\)
• D. \(\pi+3/2\)
• E. \(\sqrt{5}/2 \pi\)

Hint:

Always visualize the geometric setup. For an inscribed circle in a square, the key is realizing the diameter equals the side length. For a circumscribed circle, the key is realizing the diagonal of the square equals the diameter.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
The problem involves the relationship between a square and a circle inscribed within it. An inscribed circle is the largest possible circle that can fit inside the square, touching all four sides.
Step 2: Key Formula or Approach:
- Area of a square with side \(s\): \(A = s^2\)
- For a circle inscribed in a square, the diameter of the circle is equal to the side length of the square: \(d = s\).
- Circumference of a circle with diameter \(d\): \(C = \pi d\).
Step 3: Detailed Explanation:
We are given that the area of the square is 5.
Let the side length of the square be \(s\).
\[ A_{square} = s^2 = 5 \] To find the side length, we take the square root of the area:
\[ s = \sqrt{5} \] A circle is inscribed in this square. This means the diameter of the circle, \(d\), is equal to the side length of the square, \(s\).
\[ d = s = \sqrt{5} \] The question asks for the circumference of this circle. The formula for circumference is \(C = \pi d\).
Substituting the value of the diameter we found:
\[ C = \pi \times \sqrt{5} \] This is commonly written as \(\sqrt{5}\pi\).
Step 4: Final Answer:
The circumference of the circle is \(\sqrt{5}\pi\).