The image shows a diagram of two adjacent rectangles labeled as A and B, and you are asked to calculate their areas, the total area, and the perimeter of the combined shape. Let's work through the problem.

Step 1: Area of Rectangle (A)

From the diagram:

• The dimensions of rectangle A are $7 \, \text{cm} \times 4 \, \text{cm}$.

So, the area of rectangle A is: $\text{Area of A} = 7 \, \text{cm} \times 4 \, \text{cm} = 28 \, \text{cm}^2$

Step 2: Area of Rectangle (B)

From the diagram:

• The dimensions of rectangle B are $3 \, \text{cm} \times 6 \, \text{cm}$.

So, the area of rectangle B is: $\text{Area of B} = 3 \, \text{cm} \times 6 \, \text{cm} = 18 \, \text{cm}^2$

Step 3: Total Area

The total area of the combined shape is the sum of the areas of rectangles A and B: $\text{Total Area} = 28 \, \text{cm}^2 + 18 \, \text{cm}^2 = 46 \, \text{cm}^2$

Step 4: Perimeter

To calculate the perimeter, sum the lengths of the outer boundary of the combined shape. The lengths are:

• Top: $7 \, \text{cm}$
• Right side: $7 \, \text{cm}$
• Bottom right: $4 \, \text{cm}$
• Bottom left: $6 \, \text{cm}$
• Vertical section (between rectangles): $3 \, \text{cm}$
• Left vertical side: $1 \, \text{cm}$

So, the perimeter is: $\text{Perimeter} = 7 \, \text{cm} + 7 \, \text{cm} + 4 \, \text{cm} + 6 \, \text{cm} + 3 \, \text{cm} + 1 \, \text{cm} = 28 \, \text{cm}$

Summary:

• Area of Rectangle (A): $28 \, \text{cm}^2$
• Area of Rectangle (B): $18 \, \text{cm}^2$
• Total Area: $46 \, \text{cm}^2$
• Perimeter: $28 \, \text{cm}$

Would you like further explanation on any part of this?

Here are some follow-up questions you might consider:

1. How would the total area change if the dimensions of rectangle B were doubled?
2. If only the width of rectangle A increased by 2 cm, how would the total area and perimeter change?
3. What if both rectangles were squares? How would this affect the perimeter and area calculations?
4. How could you split the combined shape into different figures to simplify area calculation?
5. If the shape was a floor plan, how would you calculate the amount of material needed to cover it?

Tip: When calculating perimeter, always ensure all external sides of a composite figure are included, even when sides are partially shared between shapes.