Calculating the Area of a Rectangle: A Step-by-Step Guide

When faced with the task of calculating the area of a rectangle, especially when you only know the length of the diagonal and the breadth, you can use the Pythagorean theorem to find the missing length. This guide will walk you through the process, using an example—a rectangle with a diagonal length of 26 cm and a breadth of 10 cm. The area of the rectangle is crucial information in many real-world scenarios, from construction and landscaping to graphic design and engineering.

Understanding the Problem

The given information is:

The length of the diagonal of the rectangle is 26 cm. The length of the breadth of the rectangle is 10 cm.

Using the Pythagorean theorem, we can find the length of the rectangle. The formula is:

d^2 l^2 b^2

Step-by-Step Solution

Given the values: d 26 cm b 10 cm Substitute into the formula: 26^2 l^2 10^2 Calculate: 676 l^2 100 Solve for l^2 by rearranging: l^2 676 - 100 l^2 576 Take the square root to find the length (l): l sqrt{576} 24 text{ cm}

Calculating the Area

Now that we know both dimensions (length and breadth), we can calculate the area using the formula for the area of a rectangle:

A l times b

Substitute the known values:

A 24 text{ cm} times 10 text{ cm} 240 text{ cm}^2

Alternative Methods

While the Pythagorean theorem is a straightforward and robust method, there are other ways to solve this problem. Here are two alternative methods:

Method 1: Direct Calculation Using the Diagonal Length

The area can also be found directly by using the diagonal and the breadth. The formula is:

A sqrt{d^2 - b^2} times b

Substitute the given values:

A sqrt{26^2 - 10^2} times 10 sqrt{676 - 100} times 10 sqrt{576} times 10 24 times 10 240 text{ cm}^2

Method 2: Using Trigonometry

Another approach would be using trigonometric functions. The length can be found by projecting the diagonal onto the other dimension. The formula is:

A d times cos(sin^{-1}(frac{b}{d})) times b

Substitute the given values:

A 26 times cos(sin^{-1}(frac{10}{26})) times 10 26 times sqrt{1 - (frac{10}{26})^2} times 10 26 times sqrt{1 - frac{100}{676}} times 10 26 times sqrt{frac{576}{676}} times 10 26 times frac{24}{26} times 10 24 times 10 240 text{ cm}^2

Conclusion

By using these methods, we have determined that the area of the rectangle is 240 square centimeters. This method is particularly useful in scenarios where you need to find missing dimensions and areas of rectangles.

Additional Tips

If you encounter similar problems, make sure to double-check your calculations, especially the square roots and trigonometric functions, to ensure accuracy. It is also helpful to review the Pythagorean theorem and trigonometry basics, as they are foundational in many mathematical and scientific fields.

Questions

Feel free to ask questions in the comments section if you need further clarification or examples. Happy learning!