Short Answer 
The answer explains that a circle is defined by its centre and radius, and compares the areas of a square and a circle, leading to the conclusion that the side length of the square is less than the radius of the circle multiplied by the square root of œA, expressed as s < r‚aoœA.
Step 1: Understanding the Circle
A circle is defined as a shape where all points are equidistant from a central point called the centre. This distance from the centre to any point on the circle is known as the radius. Key aspects of a circle include:
- Centre: The middle point of the circle.
- Radius: The distance from the centre to the edge.
- Circle Equation: Expressed in terms of the radius and centre coordinates.
Step 2: Comparing Areas of Shapes
The area of a square with side length s is represented by the formula s¬≤, while the area of a circle with radius r is calculated as œAr¬≤. For comparison, we assume that the area of the square is less than the area of the circle:
- Area of Square: s²
- Area of Circle: œAr¬≤
- Formula for Comparison: Establish that s¬≤ < œAr¬≤.
Step 3: Deriving the Relationship
To determine the relationship between the side length of the square and the circle’s radius, we take the square root of both sides of our area inequality. This leads us to the conclusion that:
- Taking Square Roots: From s¬≤ < œAr¬≤, we get s < r‚aoœA.
- Final Conclusion: The length of the side of the square is less than the radius of the circle multiplied by the square root of œA.
- Implication: Thus, s must always be smaller than r‚aoœA.