Looking at this sixth figure, I can see a square with four circles inscribed in it. The diagram shows two circles on the bottom and two on top, arranged in a 2x2 grid within the square.
Since the problem states "Four circles inscribed inside a square" and the circles appear to be identical and touching each other and the sides of the square, I need to determine the relationship between the circle radius and the square side.
If we assume each circle has radius r, then the diameter of each circle is 2r. Since there are two circles side by side horizontally and vertically, the side of the square would be 4r.
Let me calculate the areas:
- Total area (square) = (4r)² = 16r²
- Shaded area (four circles) = 4 × πr² = 4πr²
- Unshaded area = 16r² - 4πr² = 4r²(4 - π)
- Geometric Probability = (4πr²)/(16r²) = π/4 ≈ 3.1416/4 ≈ 0.785
Now I have all six figures analyzed. Let me summarize the results:
1. Rectangle with smaller rectangles: Total area = 60, Shaded area = 26, Unshaded area = 34, Probability = 26/60 = 13/30 ≈ 0.433
2. Rectangle with diagonal shading: Total area = 1 (unit), Shaded area = 0.5, Unshaded area = 0.5, Probability = 0.5
3. Right triangle in rectangle: Total area = 200, Shaded area = 15, Unshaded area = 185, Probability = 15/200 = 3/40 = 0.075
4. Circle inscribed in square: Total area = 100, Shaded area = 25π, Unshaded area = 100 - 25π, Probability = π/4 ≈ 0.785
5. Circle divided into sectors: Total area = 1 (unit), Shaded area = 3/8, Unshaded area = 5/8, Probability = 3/8 = 0.375
6. Four circles in square: Total area = 16r², Shaded area = 4πr², Unshaded area = 4r²(4 - π), Probability = π/4 ≈ 0.785
The geometric probability for each figure is the ratio of the shaded area to the total area, which I've calculated for each case.
Parent Tip: Review the logic above to help your child master the concept of geometry probability worksheets.
