This worksheet helps students practice geometric probability by calculating the ratio of shaded area to total area for various shapes.

Here are the step-by-step solutions for the geometric probability problems on the worksheet.

1. Rectangle with a Triangle
* Total Area: The rectangle has a base of 8 and height of 5. Area = $8 \times 5 = 40$.
* Shaded Area: The shaded part is a triangle with base 8 and height 5. Area = $\frac{1}{2} \times 8 \times 5 = 20$.
* Probability: $\frac{20}{40} = \frac{1}{2}$.

2. Circle with a Sector
* Total Area: The circle has a radius of 3. Area = $\pi \times 3^2 = 9\pi$.
* Shaded Area: The shaded sector is $60^\circ$. A full circle is $360^\circ$, so the fraction is $\frac{60}{360} = \frac{1}{6}$. Shaded Area = $\frac{1}{6} \times 9\pi = 1.5\pi$.
* Probability: The ratio of the angles is simply $\frac{60}{360} = \frac{1}{6}$.

3. Octagon
* Total Parts: The octagon is divided into 8 equal triangular sections.
* Shaded Parts: 3 of these sections are shaded.
* Probability: $\frac{3}{8}$.

4. Large Triangle made of Small Triangles
* Total Parts: Counting the small triangles, there are 16 total (4 rows: 1+3+5+7).
* Shaded Parts: There are 6 shaded small triangles.
* Probability: $\frac{6}{16}$ simplifies to $\frac{3}{8}$.

5. Annulus (Ring)
* Total Area: The outer circle has radius 5. Area = $\pi \times 5^2 = 25\pi$.
* Shaded Area: This is the area of the ring. Outer Area ($25\pi$) minus Inner Area ($\pi \times 3^2 = 9\pi$). Shaded Area = $16\pi$.
* Probability: $\frac{16\pi}{25\pi} = \frac{16}{25}$.

6. Trapezoid
* Total Area: Base 1 = 6, Base 2 = 10, Height = 4. Area = $\frac{6 + 10}{2} \times 4 = 8 \times 4 = 32$.
* Shaded Area: The shaded part is a triangle with base 6 and height 4. Area = $\frac{1}{2} \times 6 \times 4 = 12$.
* Probability: $\frac{12}{32}$ simplifies to $\frac{3}{8}$.

7. Rectangle with Two Circles
* Total Area: The width is 4 (two circles of diameter 2 side-by-side). The height is 4 (diameter 2 + diameter 2). Total Area = $4 \times 4 = 16$.
* Shaded Area: Two circles with radius 1. Area of one = $\pi(1)^2 = \pi$. Two circles = $2\pi$.
* Probability: $\frac{2\pi}{16} = \frac{\pi}{8}$.

8. Rectangle with Diagonals
* Logic: The diagonals of a rectangle divide it into 4 triangles of equal area.
* Shaded Parts: 2 of the 4 triangles are shaded.
* Probability: $\frac{2}{4} = \frac{1}{2}$.

9. Circle with Inscribed Square
* Total Area: Radius is 4. Area = $\pi \times 4^2 = 16\pi$.
* Shaded Area: The square's diagonal is the diameter of the circle ($4+4=8$). Area of a square using diagonal $d$ is $\frac{d^2}{2}$. Area = $\frac{8^2}{2} = \frac{64}{2} = 32$.
* Probability: $\frac{32}{16\pi} = \frac{2}{\pi}$.

10. Diamond (Square) with Internal Shapes
* Total Area: The diagonal is given as 8. Area = $\frac{8^2}{2} = \frac{64}{2} = 32$.
* Shaded Area:
* Top Triangle: Base 4, Height 4 (half the diagonal). Area = $\frac{1}{2} \times 4 \times 4 = 8$.
* Bottom Right Triangle: Base 4, Height 4. Area = $\frac{1}{2} \times 4 \times 4 = 8$.
* Total Shaded = $8 + 8 = 16$.
* Probability: $\frac{16}{32} = \frac{1}{2}$.

11. Square with Circles
* Total Area: Side length is 8. Area = $8 \times 8 = 64$.
* Shaded Area:
* Large Circle: Radius 2. Area = $4\pi$.
* Two Small Circles: Diameter 4 means Radius 2. Area of one = $4\pi$. Two of them = $8\pi$.
* Total Shaded = $4\pi + 8\pi = 12\pi$.
* Probability: $\frac{12\pi}{64} = \frac{3\pi}{16}$.

12. Rectangle with Circles
* Total Area: Width is 4 (diameter of the large circle). Height is 8 (two diameters stacked). Area = $4 \times 8 = 32$.
* Shaded Area:
* Top Circle: Radius 2. Area = $4\pi$.
* Bottom Semicircles: Two semicircles of radius 2 make one full circle. Area = $4\pi$.
* Total Shaded = $4\pi + 4\pi = 8\pi$.
* Probability: $\frac{8\pi}{32} = \frac{\pi}{4}$.

Final Answer:
1. 1/2
2. 1/6
3. 3/8
4. 3/8
5. 16/25
6. 3/8
7. π/8
8. 1/2
9. 2/π
10. 1/2
11. 3π/16
12. π/4