A square is inscribed in a circle. If the area of the shaded region is 224 cm², calculate the radius.

To find the radius of a circle in which a square is inscribed, we first need to understand the relationship between the square and the circle.

Understanding the Shapes

When a square is inscribed in a circle, the diameter of the circle is equal to the diagonal of the square. If we denote the side length of the square as s, the diagonal can be calculated using the formula:

• Diagonal = s√2

Area of the Circle and Square

The area of the shaded region (the area of the circle minus the area of the square) is given as 224 cm². The formulas for the areas are:

• Area of the circle = πr²
• Area of the square = s²

Setting Up the Equation

We can express the area of the shaded region as:

Area of the shaded region = Area of the circle - Area of the square

Substituting the formulas, we have:

224 = πr² - s²

Relating Side Length and Radius

Since the diagonal of the square is equal to the diameter of the circle, we can express the radius in terms of the side length:

r = (s√2) / 2

Squaring both sides gives:

r² = (s² / 2)

Substituting and Solving

Now, we can substitute r² in our shaded area equation:

224 = π(2s²) - s²

This simplifies to:

224 = (2π - 1)s²

Now, solving for s²:

s² = 224 / (2π - 1)

Finding the Radius

Next, we substitute s² back into the equation for r²:

r² = (s² / 2) = 112 / (2π - 1)

Finally, to find the radius:

r = √(112 / (2π - 1))

Final Calculation

Using the approximate value of π (3.14) for calculation:

r ≈ √(112 / (2 * 3.14 - 1))

Calculating this gives:

r ≈ √(112 / 5.28) ≈ √21.18 ≈ 4.6 cm

Thus, the radius of the circle is approximately 4.6 cm.