To double the area of a square, the side length of the new square must be precisely the side length of the original square multiplied by the square root of 2 (approximately 1.414). This new side length is equivalent to the length of the diagonal of the original square.
Understanding Area Doubling
A square's area is calculated by squaring its side length ($s^2$). To double this area, we need a new square with an area of $2s^2$. If the side length of this new square is $s'$, then $s'^2 = 2s^2$. Solving for $s'$, we find that $s' = s \times \sqrt{2}$.
The value $\sqrt{2}$ is an irrational number, approximately 1.41421356. This means the side of the new square must be about 1.414 times longer than the side of the original square.
The Key Connection: The Diagonal
Remarkably, the length of the diagonal of any square is exactly its side length multiplied by the square root of 2. For an original square with side length $s$, its diagonal measures $s \times \sqrt{2}$. Therefore, to double the area of a square, you simply need to construct a new square whose side length is equal to the diagonal of the original square.
This relationship provides a straightforward method for doubling a square's area, whether through calculation or geometric construction.
Methods to Double a Square's Area
Here are practical approaches to double the area of a square:
1. Calculation and Construction Method
This method is suitable when you know the original square's dimensions and can use a ruler or precise drawing tools.
- Measure the original side: Determine the side length ($s$) of your initial square.
- Calculate the new side length: Multiply the original side length by $\sqrt{2}$.
- Example: If the original square has a side length of 5 cm, the new side length will be $5 \text{ cm} \times \sqrt{2} \approx 5 \text{ cm} \times 1.414 = 7.07 \text{ cm}$.
- Draw the new square: Construct a new square using this calculated side length.
2. Geometric Construction Using the Diagonal
This elegant method uses the original square's properties directly, eliminating the need for precise calculations involving $\sqrt{2}$.
- Draw the original square: Start with your initial square, let's call it Square A.
- Draw a diagonal: Draw one of the diagonals of Square A (e.g., from one corner to the opposite corner).
- Use the diagonal as the new side: This diagonal is the exact length required for the side of your new, larger square. Construct a new square (Square B) where each of its sides is equal to the length of the diagonal you just drew from Square A.
3. Visual Midpoint Construction
This method provides a visual demonstration of the doubling principle and is useful for understanding the geometry.
- Create a larger square: Arrange four identical copies of your original square (Square A) to form a larger $2 \times 2$ grid. This creates a larger square with a side length of $2s$.
- Connect the midpoints: Find the midpoint of each of the four outer sides of this large $2s \times 2s$ square.
- Form the doubled square: Connect these four midpoints. The shape formed in the center is a square, and its area will be exactly double that of your original Square A. Each side of this inner square is the hypotenuse of a right triangle with legs of length $s$, thus having a length of $s \times \sqrt{2}$.
Comparing Square Properties
| Property | Original Square | Doubled Area Square |
|---|---|---|
| Side Length | $s$ | $s \times \sqrt{2}$ |
| Area | $s^2$ | $2s^2$ |
| Perimeter | $4s$ | $4s \times \sqrt{2}$ |
| Diagonal Length | $s \times \sqrt{2}$ | $2s$ |
For more general information on the properties of squares in geometry, you can refer to resources like Wikipedia's article on the square.
Understanding how to double the area of a square has practical applications in fields such as architecture, design, and land management, where scaling dimensions while maintaining proportional area is crucial.
