Calculating the Area of a Square Given Its Diagonal

Mathematics is a powerful tool in understanding geometric shapes and their properties. One common problem in geometry is finding the area of a square when the length of its diagonal is known. This guide will explore the methods to solve such a problem and provide a detailed step-by-step solution.

Understanding the Problem

The problem is to find the area of a square whose diagonal length is 2.4 meters. This involves understanding the relationship between the diagonal and the sides of the square. The relationship is given by the formula:

d s√2

where (d) is the diagonal and (s) is the side length of the square.

Solving the Problem Using the Pythagorean Theorem

To solve the problem, we start by expressing the side length in terms of the diagonal:

Given (d 2.4) meters, the side length (s) can be found using the formula: [s frac{d}{sqrt{2}}] Substituting the diagonal value: [s frac{2.4}{sqrt{2}}] Multiplying the numerator by (sqrt{2}) and keeping the denominator as 2: [s frac{2.4 cdot sqrt{2}}{2} 1.2sqrt{2}] meters

Now that we have the side length of the square, we can use the formula for the area of a square:

A s^2

Substituting (s 1.2sqrt{2}) into the formula:

[A (1.2sqrt{2})^2] [A 1.44 cdot 2 2.88] square meters

Thus, the area of the square is 2.88 square meters.

Variations in Calculating the Area

There are multiple methods to solve the same problem. Here are a few alternative approaches:

Using the Relationship Between Diagonal and Area

Another way to solve this problem is by using the relationship involving the diagonal directly. We know:

Area (frac{d^2}{2})

Given (d 2.4) meters:

[Area frac{2.4^2}{2}] [Area frac{5.76}{2} 2.88] square meters

Using the Pythagorean Theorem in a Different Form

Given that the diagonal of a square is the hypotenuse of a right triangle formed by two sides of the square, we can use the Pythagorean theorem:

2s^2 d^2

Given (d 2.4) meters:

[2s^2 2.4^2] [2s^2 5.76] [s^2 2.88] [s sqrt{2.88}] [s 1.2sqrt{2}] meters [Area s^2 2.88] square meters

Conclusion

The area of a square with a diagonal of 2.4 meters is 2.88 square meters. This can be calculated using various methods, all of which rely on the fundamental relationships in geometry. By understanding and applying these principles, you can solve similar problems effectively.