To solve the problem where a square is inscribed in a circle with a given radius, let's start by analyzing the relationship between the radius of the circle and the side length of the square.

Step 1: Understand the Geometry

• A square is inscribed in a circle, which means the circle is the circumscribed circle around the square.
• The radius of the circle (r) is related to the diagonal of the square.
• If the side length of the square is s, the diagonal of the square can be calculated using the Pythagorean theorem.

The diagonal of a square (d) in terms of its side length (s) is: $d = s\sqrt{2}$

Step 2: Relate Radius to Diagonal

• Since the square is inscribed in the circle, the diagonal of the square is equal to the diameter of the circle.
• The diameter of the circle is twice the radius (r), so: $d = 2r$

Step 3: Set Up the Relationship

We know the diagonal in terms of the side length of the square is s√2, and it is also 2r in terms of the radius of the circle. Therefore: $s\sqrt{2} = 2r$

Step 4: Plug in the Given Radius

The radius of the circle is given as: $r = \frac{20\sqrt{2}}{2} = 10\sqrt{2}$

Step 5: Solve for Side Length

Substituting r into the equation: $s\sqrt{2} = 2 \times 10\sqrt{2}$ $s\sqrt{2} = 20\sqrt{2}$

To solve for s, divide both sides by \sqrt{2}: $s = \frac{20\sqrt{2}}{\sqrt{2}}$ $s = 20$

Final Answer

The side length of the square is 20 inches.

Would you like me to explain any particular step in more detail, or do you have any questions?

Here are some related questions you might find interesting:

1. How would you calculate the area of the square inscribed in the circle?
2. What is the circumference of the circle if its radius is $10\sqrt{2}$?
3. How would the problem change if the square were circumscribing the circle instead?
4. What is the relationship between the area of the circle and the area of the inscribed square?
5. How can you derive the formula for the diagonal of a square from its side length?

Tip: When dealing with inscribed shapes, always try to visualize or draw the relationships between radius, diameter, and diagonals to make solving easier.