ABCD is a square with sides 12 cm each. Find its perimeter and area. Cut off a rectangle measuring C 8cm 2 cm by 4 cm from a corner as shown in the figure. Find the perimeter and area of the new polygon thus formed.

Answer

To solve the problem step by step, we will first find the perimeter and area of the square ABCD, and then we will calculate the perimeter and area of the new polygon after cutting off the rectangle.

### Step 1: Find the Perimeter of Square ABCD
The formula for the perimeter \( P \) of a square is given by:
\[ P = 4 \times \text{side} \]
Given that each side of the square ABCD is 12 cm:
\[ P = 4 \times 12 = 48 \text{ cm} \]

### Step 2: Find the Area of Square ABCD
The formula for the area \( A \) of a square is given by:
\[ A = \text{side}^2 \]
Using the side length of 12 cm:
\[ A = 12^2 = 144 \text{ cm}^2 \]

### Step 3: Find the Area of the Rectangle Cut Off
The dimensions of the rectangle are given as 2 cm by 4 cm. The area \( A_r \) of the rectangle can be calculated using the formula:
\[ A_r = \text{length} \times \text{breadth} \]
Thus,
\[ A_r = 2 \times 4 = 8 \text{ cm}^2 \]

### Step 4: Find the Area of the New Polygon
To find the area of the new polygon formed after cutting off the rectangle, we subtract the area of the rectangle from the area of the square:
\[ A_{\text{new}} = A - A_r \]
Substituting the values we found:
\[ A_{\text{new}} = 144 - 8 = 136 \text{ cm}^2 \]

### Step 5: Find the Perimeter of the New Polygon
To find the perimeter of the new polygon, we need to consider the perimeter of the square and subtract the lengths of the sides of the rectangle that were removed. The rectangle has one side of 4 cm and another side of 2 cm, so we remove these lengths from the perimeter of the square:
\[ P_{\text{new}} = P - (4 + 2) \]
Substituting the values:
\[ P_{\text{new}} = 48 - 6 = 42 \text{ cm} \]

### Summary of Results
- Perimeter of square ABCD: **48 cm**
- Area of square ABCD: **144 cm²**
- Area of the rectangle cut off: **8 cm²**
- Area of the new polygon: **136 cm²**
- Perimeter of the new polygon: **42 cm**