The area of a square will be increased by a factor of 9 if its side is tripled.
Understanding the Change in Area
When the side of a square is tripled, its new area becomes nine times its original area. This is a fundamental principle of geometric scaling.
Let's illustrate with a simple breakdown:
- Original Side: Let
srepresent the length of one side of the square. - Original Area: The area of the square is calculated as
side × side, which iss². - New Side: If the side is tripled, the new side length becomes
3s. - New Area: To find the new area, we square the new side length:
(3s)².(3s)² = 3s × 3s(3s)² = 9s²
As you can see, the new area (9s²) is exactly nine times the original area (s²).
Practical Example
Consider a square with an initial side length and observe how its area changes when the side is tripled.
| Property | Original Square | Tripled Side Square |
|---|---|---|
| Side Length | 2 units | 6 units (2 × 3) |
| Area | 2² = 4 sq units | 6² = 36 sq units |
In this example, the area increased from 4 square units to 36 square units.
36 ÷ 4 = 9. This confirms that the area is increased by a factor of 9.
Why Does It Work This Way?
This relationship holds true because the area is a two-dimensional measurement. When you multiply the length of a side by a factor, you are essentially applying that factor to both dimensions (length and width). If each dimension is multiplied by 3, the overall area is multiplied by 3 × 3, which equals 9.
For more information on calculating the area of squares and other geometric shapes, you can refer to resources like Khan Academy's Geometry section.
