GCSE maths worksheet: area and perimeter of compound shapes ... - Free Printable

Problem: Compound Shapes - Area & Perimeter

We are tasked with finding the perimeter and area of compound shapes. Let's solve each shape step by step.

---

#### Shape 1:
The shape is a rectangle with a smaller rectangle cut out from one corner.

- Dimensions:
- Larger rectangle: \( 15 \, \text{cm} \times 9 \, \text{cm} \)
- Smaller rectangle (cut-out): \( 3 \, \text{cm} \times 11 \, \text{cm} \)

##### Step 1: Perimeter
The perimeter of the compound shape is the sum of all its outer edges. Since the smaller rectangle is cut out, we need to account for the remaining outer edges.

- The larger rectangle has dimensions \( 15 \, \text{cm} \) and \( 9 \, \text{cm} \).
- The smaller rectangle removes part of the perimeter, but it also adds new edges where the cut is made.

To calculate the perimeter:
1. Calculate the perimeter of the larger rectangle:
\[
P_{\text{large}} = 2 \times (15 + 9) = 2 \times 24 = 48 \, \text{cm}
\]
2. Subtract the lengths of the sides that are no longer part of the perimeter (the two sides of the smaller rectangle that were part of the larger rectangle):
\[
\text{Removed length} = 11 + 3 = 14 \, \text{cm}
\]
3. Add the new edges created by the cut (the two sides of the smaller rectangle that are now exposed):
\[
\text{Added length} = 11 + 3 = 14 \, \text{cm}
\]
4. The total perimeter remains unchanged:
\[
P_{\text{total}} = 48 \, \text{cm}
\]

##### Step 2: Area
The area of the compound shape is the area of the larger rectangle minus the area of the smaller rectangle.

1. Calculate the area of the larger rectangle:
\[
A_{\text{large}} = 15 \times 9 = 135 \, \text{cm}^2
\]
2. Calculate the area of the smaller rectangle:
\[
A_{\text{small}} = 3 \times 11 = 33 \, \text{cm}^2
\]
3. Subtract the area of the smaller rectangle from the area of the larger rectangle:
\[
A_{\text{total}} = 135 - 33 = 102 \, \text{cm}^2
\]

##### Final Answer for Shape 1:
\[
\boxed{48 \, \text{cm}, 102 \, \text{cm}^2}
\]

---

#### Shape 2:
The shape is a rectangle with a smaller square cut out from one corner.

- Dimensions:
- Larger rectangle: \( 15 \, \text{cm} \times 13 \, \text{cm} \)
- Smaller square (cut-out): \( 4 \, \text{cm} \times 4 \, \text{cm} \)

##### Step 1: Perimeter
The perimeter of the compound shape is the sum of all its outer edges. Since the smaller square is cut out, we need to account for the remaining outer edges.

- The larger rectangle has dimensions \( 15 \, \text{cm} \) and \( 13 \, \text{cm} \).
- The smaller square removes part of the perimeter, but it also adds new edges where the cut is made.

To calculate the perimeter:
1. Calculate the perimeter of the larger rectangle:
\[
P_{\text{large}} = 2 \times (15 + 13) = 2 \times 28 = 56 \, \text{cm}
\]
2. Subtract the lengths of the sides that are no longer part of the perimeter (the two sides of the smaller square that were part of the larger rectangle):
\[
\text{Removed length} = 4 + 4 = 8 \, \text{cm}
\]
3. Add the new edges created by the cut (the two sides of the smaller square that are now exposed):
\[
\text{Added length} = 4 + 4 = 8 \, \text{cm}
\]
4. The total perimeter remains unchanged:
\[
P_{\text{total}} = 56 \, \text{cm}
\]

##### Step 2: Area
The area of the compound shape is the area of the larger rectangle minus the area of the smaller square.

1. Calculate the area of the larger rectangle:
\[
A_{\text{large}} = 15 \times 13 = 195 \, \text{cm}^2
\]
2. Calculate the area of the smaller square:
\[
A_{\text{small}} = 4 \times 4 = 16 \, \text{cm}^2
\]
3. Subtract the area of the smaller square from the area of the larger rectangle:
\[
A_{\text{total}} = 195 - 16 = 179 \, \text{cm}^2
\]

##### Final Answer for Shape 2:
\[
\boxed{56 \, \text{cm}, 179 \, \text{cm}^2}
\]

---

Summary of Answers:

1. Shape 1: Perimeter = \( 48 \, \text{cm} \), Area = \( 102 \, \text{cm}^2 \)
2. Shape 2: Perimeter = \( 56 \, \text{cm} \), Area = \( 179 \, \text{cm}^2 \)

If you have more shapes to solve, please provide them, and I will continue the process!