A path of uniform width, 3 m, runs around the outside of a square field of side 21 m. Find the area of the path.

Answer

To find the area of the path that runs around the outside of a square field, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the dimensions of the square field**:
- The side length of the square field is given as 21 m.

2. **Determine the width of the path**:
- The width of the path surrounding the square field is given as 3 m.

3. **Calculate the dimensions of the larger square (including the path)**:
- The total side length of the larger square (which includes the path) can be calculated by adding the width of the path on both sides of the square field:
\[ \text{Total side length} = \text{Side length of square field} + 2 \times \text{Width of path} \]
\[ \text{Total side length} = 21 \, \text{m} + 2 \times 3 \, \text{m} = 21 \, \text{m} + 6 \, \text{m} = 27 \, \text{m} \]

4. **Calculate the area of the larger square**:
- The area of the larger square (ABCD) can be calculated using the formula for the area of a square, which is side length squared:
\[ \text{Area of larger square} = (\text{Total side length})^2 = (27 \, \text{m})^2 = 729 \, \text{m}^2 \]

5. **Calculate the area of the smaller square (the field)**:
- The area of the smaller square (PQRS) can also be calculated:
\[ \text{Area of smaller square} = (\text{Side length of square field})^2 = (21 \, \text{m})^2 = 441 \, \text{m}^2 \]

6. **Find the area of the path**:
- The area of the path is the difference between the area of the larger square and the area of the smaller square:
\[ \text{Area of the path} = \text{Area of larger square} - \text{Area of smaller square} \]
\[ \text{Area of the path} = 729 \, \text{m}^2 - 441 \, \text{m}^2 = 288 \, \text{m}^2 \]

### Final Answer:
The area of the path is **288 m²**.