To find the area of the shaded regions, we need to calculate the area of the main outer shape and subtract the area of the unshaded (white) inner shape.
Here is the step-by-step solution for each problem:
1. Rectangle with a Triangle cut out
*
Outer Shape (Rectangle): Area = width × height = $20 \text{ in} \times 8 \text{ in} = 160 \text{ in}^2$.
*
Inner Shape (Triangle): The base of the triangle is found by subtracting the side segments from the total width: $20 - 5.5 - 5.5 = 9 \text{ in}$. The height is given as $4 \text{ in}$.
* Area = $\frac{1}{2} \times \text{base} \times \text{height} = 0.5 \times 9 \times 4 = 18 \text{ in}^2$.
*
Shaded Area: $160 - 18 = 142 \text{ in}^2$.
2. Parallelogram with a Triangle cut out
*
Outer Shape (Parallelogram): Area = base × height = $15 \text{ ft} \times 9 \text{ ft} = 135 \text{ ft}^2$.
*
Inner Shape (Triangle): Base = $3 \text{ ft}$, Height = $4 \text{ ft}$.
* Area = $\frac{1}{2} \times 3 \times 4 = 6 \text{ ft}^2$.
*
Shaded Area: $135 - 6 = 129 \text{ ft}^2$.
3. Triangle with a Circle cut out
*
Outer Shape (Triangle): Area = $\frac{1}{2} \times \text{base} \times \text{height} = 0.5 \times 12 \text{ yd} \times 14 \text{ yd} = 84 \text{ yd}^2$.
*
Inner Shape (Circle): Radius ($r$) = $3 \text{ yd}$. Using $\pi \approx 3.14$:
* Area = $\pi r^2 = 3.14 \times 3^2 = 3.14 \times 9 = 28.26 \text{ yd}^2$.
*
Shaded Area: $84 - 28.26 = 55.74 \text{ yd}^2$.
4. Circle with a Trapezoid cut out
*
Outer Shape (Circle): Diameter is $20 \text{ m}$, so radius ($r$) is $10 \text{ m}$.
* Area = $\pi r^2 = 3.14 \times 10^2 = 3.14 \times 100 = 314 \text{ m}^2$.
*
Inner Shape (Trapezoid): Height ($h$) = $2 \text{ m}$. Parallel sides are $4 \text{ m}$ and $8 \text{ m}$.
* Area = $\frac{(b_1 + b_2)}{2} \times h = \frac{(4 + 8)}{2} \times 2 = 6 \times 2 = 12 \text{ m}^2$.
*
Shaded Area: $314 - 12 = 302 \text{ m}^2$.
5. Square with a Parallelogram cut out
*
Outer Shape (Square): Area = side × side = $22 \text{ cm} \times 22 \text{ cm} = 484 \text{ cm}^2$.
*
Inner Shape (Parallelogram): Area = base × height = $10 \text{ cm} \times 8 \text{ cm} = 80 \text{ cm}^2$.
*
Shaded Area: $484 - 80 = 404 \text{ cm}^2$.
6. Rectangle with a Triangle cut out
*
Outer Shape (Rectangle): Area = $25 \text{ ft} \times 10 \text{ ft} = 250 \text{ ft}^2$.
*
Inner Shape (Triangle): The triangle is cut into the side. Its base corresponds to the height of the rectangle ($10 \text{ ft}$) and its height (depth into the rectangle) is $4.5 \text{ ft}$.
* Area = $\frac{1}{2} \times 10 \times 4.5 = 22.5 \text{ ft}^2$.
*
Shaded Area: $250 - 22.5 = 227.5 \text{ ft}^2$.
Final Answer:
1. 142 in²
2. 129 ft²
3. 55.74 yd²
4. 302 m²
5. 404 cm²
6. 227.5 ft²
Parent Tip: Review the logic above to help your child master the concept of compound shapes worksheet with answers.
