Calculating the Total Area of the Square Park and Garden

The problem asks us to find the total area of a square-shaped park when a larger square-shaped garden is developed around it. We are given the side length of the inner park and the side length of the larger square formed by the park and the garden combined.

Here's what we know:

• The park is square-shaped with a side of 12 m.
• A square-shaped garden is developed around the park.
• The combined area (park plus garden) forms a larger square with a side of 24 m.

We need to find the total area of this larger square, which includes both the park and the garden.

Understanding Area of a Square

The area of any square is calculated by multiplying its side length by itself. The formula is:

\( \text{Area} = \text{side} \times \text{side} = \text{side}^2 \)

The unit for area is typically square meters (\( \text{m}^2 \)) if the side is measured in meters.

Step-by-Step Calculation of Total Area

The question states that the square-shaped garden developed around the park results in a larger square with a side of 24 m. This 24 m side represents the total length of the outer square, which encompasses both the inner park and the surrounding garden.

So, the side length of the total area (park + garden) is 24 m.

Now, we can calculate the total area using the area formula:

Side length of the total area = 24 m

Total Area = \( (\text{Side length})^2 \)

Total Area = \( (24 \text{ m})^2 \)

Total Area = \( 24 \times 24 \text{ m}^2 \)

Let's perform the multiplication:

So, the total area is 576 square meters.

Total Area = \( 576 \text{ m}^2 \)

This is the area of the larger square that includes the park and the garden.

Let's quickly verify the options provided:

• Option 1: \( 324 \text{ m}^2 \)
• Option 2: \( 576 \text{ m}^2 \)
• Option 3: \( 288 \text{ m}^2 \)
• Option 4: \( 144 \text{ m}^2 \) (This is the area of the park only: \( 12 \times 12 \text{ m}^2 \))

Our calculated total area matches Option 2.

Summary of the Calculation

• Identify the side length of the large square formed by the park and garden. This is given as 24 m.
• Use the formula for the area of a square: Area = side \(\times\) side.
• Calculate \( 24 \text{ m} \times 24 \text{ m} = 576 \text{ m}^2 \).
• This result, \( 576 \text{ m}^2 \), is the total area including the park and the garden.

The total area of the park including the garden is \( 576 \text{ m}^2 \).

Revision Table: Key Concepts for Area Calculation

Additional Information: Area of the Garden Only

The question asks for the total area (park + garden). However, sometimes you might be asked to find the area of the garden only.

To find the area of the garden only, you would subtract the area of the park from the total area.

• Area of the park = side \(\times\) side = \( 12 \text{ m} \times 12 \text{ m} = 144 \text{ m}^2 \).
• Total area (park + garden) = \( 576 \text{ m}^2 \) (as calculated above).
• Area of the garden only = Total area - Area of the park
• Area of the garden only = \( 576 \text{ m}^2 - 144 \text{ m}^2 \)
• Area of the garden only = \( 432 \text{ m}^2 \)

This shows that the garden itself has an area of \( 432 \text{ m}^2 \), and when added to the park's area (\( 144 \text{ m}^2 \)), you get the total area of \( 576 \text{ m}^2 \).