Calculating the Remaining Area After Cut-Out Squares

Understanding the concept of area calculation in geometry is crucial for various practical applications in mathematics, engineering, and design. A common problem that often arises is determining the remaining area when squares are cut out of a larger square. This article will walk you through the calculation process using an example where 4 squares, each with a side of 3 cm, are cut from the corners of a square-shaped paper with an area of 576 cm2.

Initial Setup and Given Information

The square-shaped paper has an area of 576 cm2. This implies that the side length of the square is √576 24 cm.

Calculating the Area of the Cut-Out Squares

The side length of each of the 4 cut-out squares is 3 cm. Therefore, the area of one of these squares can be calculated as:

3 cm × 3 cm 9 cm2

Since there are 4 such squares, the total area removed is:

4 × 9 cm2 36 cm2

Calculating the Remaining Area

To find the area of the remaining paper, we need to subtract the total area of the cut-out squares from the original area of the square paper:

576 cm2 - 36 cm2 540 cm2

Alternative Calculation Method

The area of one of the removed squares is calculated as 3 cm x 3 cm 9 cm2. Cutting 4 such squares removes a total area of 4 x 9 cm2 36 cm2. The remaining area can therefore be calculated as:

576 cm2 - 36 cm2 540 cm2

This can also be verified by considering the larger square as being composed of 64 smaller squares, each with a side of 3 cm:

The side of the paper corresponds to 8 times the side of the removed square. Therefore, the paper is made up of 8 x 8 64 smaller squares. Removing 4 corner squares leaves 64 - 4 60 smaller squares. Each of these smaller squares has an area of 3 cm x 3 cm 9 cm2, so the remaining area is:

60 x 9 cm2 540 cm2

Conclusion

Through both direct and indirect methods, we have determined that the remaining area of the paper after cutting out 4 squares, each with a side of 3 cm, from each corner of a square-shaped paper with an area of 576 cm2, is 540 cm2.

Understanding such calculations can be of great help in various fields, including design, construction, and engineering. By mastering these fundamental concepts, you can solve a wide range of real-world problems.