Finding The Side Length Of A Square With 450 Square Inches Area

Determining the side length of a square given its area is a fundamental concept in geometry. In this article, we will explore how to calculate the side length of a square with an area of 450 square inches. This problem involves understanding the relationship between the area and side length of a square and applying basic algebraic principles.

Understanding the Relationship Between Area and Side Length

The area of a square is calculated by squaring the length of one of its sides. Mathematically, this can be represented as:

Area = side × side = side²

Conversely, if we know the area of a square, we can find the side length by taking the square root of the area. This can be expressed as:

Side = √Area

This relationship forms the basis for solving our problem. By applying this understanding, we can accurately determine the side length of a square with a given area.

Calculating the Side Length

To find the side length of a square with an area of 450 square inches, we need to apply the formula mentioned above:

Side = √Area

In this case, the area is 450 square inches. Therefore, we have:

Side = √450

Now, we need to simplify the square root of 450. To do this, we look for perfect square factors of 450. A perfect square is a number that can be obtained by squaring an integer (e.g., 4, 9, 16, 25, etc.).

We can factor 450 as follows:

450 = 2 × 225

Notice that 225 is a perfect square (15² = 225). We can further break down 225:

225 = 15 × 15

So, we can rewrite 450 as:

450 = 2 × 15 × 15 = 15² × 2

Now, we can substitute this back into our equation for the side length:

Side = √450 = √(15² × 2)

Using the property of square roots that √(a × b) = √a × √b, we can simplify this as:

Side = √15² × √2

Since the square root of 15² is 15, we have:

Side = 15√2 inches

Therefore, the side length of a square with an area of 450 square inches is 15√2 inches. This calculation demonstrates the practical application of understanding the relationship between the area and side length of a square, as well as the simplification of square roots.

Analyzing the Given Options

Now, let's analyze the given options to determine which one represents the side length of the square with an area of 450 square inches:

1. 15√2 inches: As we calculated above, this is the correct side length.
2. 15√3 inches: This is incorrect. If we were to square this value, we would get (15√3)² = 15² × (√3)² = 225 × 3 = 675 square inches, which is not equal to 450 square inches.
3. 112.5 inches: This is incorrect. If this were the side length, the area would be 112.5² = 12656.25 square inches, which is far from 450 square inches.
4. 115.5 inches: This is also incorrect. Squaring this value gives us 115.5² = 13340.25 square inches, which is not equal to 450 square inches.

Detailed Explanation of Incorrect Options

To further clarify why the other options are incorrect, let's delve into the calculations:

• 15√3 inches: To understand why this is incorrect, we square it to find the area. (15√3)² equals 15² multiplied by (√3)². This simplifies to 225 multiplied by 3, resulting in an area of 675 square inches. This value is significantly higher than the given area of 450 square inches. The presence of √3 instead of √2 in the side length expression leads to a larger area when squared, making it an incorrect answer. The correct side length involves √2 because the factorization of 450 includes a factor of 2 under the square root.
• 112.5 inches: This option can be quickly dismissed by squaring it to find the area. 112. 5² equals 12656.25 square inches, a value much larger than the target area of 450 square inches. This suggests that 112.5 inches is an overestimate of the actual side length. The side length must be a value that, when squared, yields 450, and 112.5 inches clearly exceeds this requirement.
• 115.5 inches: Similar to the previous option, squaring 115.5 inches gives an area that is far too large. 115. 5² equals 13340.25 square inches, which is significantly greater than 450 square inches. This indicates that 115.5 inches is also an overestimate of the side length. The correct side length must be considerably smaller to produce the given area of 450 square inches. These large values highlight the importance of understanding the scale and relationships between side length and area in geometric calculations.

Therefore, the only correct option is 15√2 inches, which we derived by correctly applying the formula and simplifying the square root. This detailed analysis not only confirms the correct answer but also illustrates why the other options are not viable, reinforcing the mathematical principles involved.

Conclusion

In conclusion, the side length of a square with an area of 450 square inches is 15√2 inches. This solution is obtained by understanding the relationship between the area and side length of a square and applying the appropriate algebraic manipulations. The other options provided are incorrect as they do not satisfy the condition that the square of the side length must equal the area of 450 square inches. This problem highlights the importance of accurately applying mathematical formulas and simplifying expressions to arrive at the correct solution.

Understanding these principles is crucial for solving a variety of geometry problems and for building a strong foundation in mathematics.