Properties of the Original Shape

Oct 13, 2025 by windailys.com 5 views

Introduction

In geometry, cutting identical squares from the corners of a rectangular sheet of paper is a common problem that requires a deep understanding of shapes and their properties. The problem presented here involves cutting squares from the corners of a rectangular sheet of paper, resulting in a new shape with a different area. In this article, we will explore the concept of cutting corners and find the area of the resultant paper.

Understanding the Problem

The problem states that Jay cuts identical squares from the corners of a rectangular sheet of paper. This means that the squares cut from the corners are of equal size and shape. The resulting shape will have a different area than the original rectangular sheet of paper. To find the area of the resultant paper, we need to understand the properties of the original shape and the shape of the cut squares.

Properties of the Original Shape

The original shape is a rectangular sheet of paper with length l and width w. The area of the original shape is given by the formula:

Area = l * w

Properties of the Cut Squares

The cut squares are identical and have a side length of x. The area of each cut square is given by the formula:

Area = x^2

Finding the Area of the Resultant Paper

To find the area of the resultant paper, we need to subtract the area of the cut squares from the area of the original shape. The area of the cut squares is equal to the sum of the areas of the individual squares. Since the squares are identical, we can multiply the area of one square by the number of squares cut.

Area of cut squares = x^2 * 4

The area of the resultant paper is given by the formula:

Area = l * w - x^2 * 4

Simplifying the Formula

We can simplify the formula by factoring out the common term x^2:

Area = l * w - 4x^2

Example Problem

Let's consider an example problem where the original shape is a rectangular sheet of paper with length l = 10 and width w = 5. The cut squares have a side length of x = 2. We can plug these values into the formula to find the area of the resultant paper:

Area = 10 * 5 - 4 * 2^2 Area = 50 - 16 Area = 34

Conclusion

In this article, we explored the concept of cutting corners from a rectangular sheet of paper and found the area of the resultant paper. We used the properties of the original shape and the shape of the cut squares to derive a formula for the area of the resultant paper. The formula involves subtracting the area of the cut squares from the area of the original shape. We also simplified the formula by factoring out the common term x^2. Finally, we solved an example problem to demonstrate the application of the formula.

Properties of the Resultant Shape

The resultant shape is a new shape with a different area than the original rectangular sheet of paper. The shape of the resultant paper is a rectangle with a length of l - 2x and a width of w - 2x. The area of the resultant paper is given by the formula:

Area = (l - 2x) * (w - 2x)

Simplifying the Formula

We can simplify the formula by expanding the product:

Area = l * w - 2lx - 2wx + 4x^2

Comparing the Formulas

We can compare the two formulas for the area of the resultant paper:

Area = l * w - 4x^2 Area = l * w - 2lx - 2wx + 4x^2

The two formulas are equivalent, but the second formula is more general and can be used to find the area of the resultant paper for any values of l, w, and x.

Real-World Applications

The problem of cutting corners from a rectangular sheet of paper has many real-world applications. For example, in architecture, designers may need to cut corners from a rectangular sheet of paper to create a new shape for a building or a structure. In engineering, engineers may need to cut corners from a rectangular sheet of paper to create a new shape for a machine or a device. In art, artists may need to cut corners from a rectangular sheet of paper to create a new shape for a sculpture or a installation.

Conclusion

Q: What is the formula for the area of the resultant paper?

A: The formula for the area of the resultant paper is:

Area = l * w - 4x^2

where l is the length of the original rectangular sheet of paper, w is the width of the original rectangular sheet of paper, and x is the side length of the cut squares.

Q: How do I find the area of the resultant paper if I don't know the side length of the cut squares?

A: If you don't know the side length of the cut squares, you can use the formula:

Area = l * w - 4x^2

However, you will need to know the side length of the cut squares to plug into the formula. If you don't know the side length of the cut squares, you can try to find it by measuring the shape of the resultant paper.

Q: Can I use the formula to find the area of the resultant paper if the cut squares are not identical?

A: No, the formula we derived assumes that the cut squares are identical. If the cut squares are not identical, you will need to use a different formula to find the area of the resultant paper.

Q: How do I find the area of the resultant paper if the original rectangular sheet of paper is not a rectangle?

A: If the original rectangular sheet of paper is not a rectangle, you will need to use a different formula to find the area of the resultant paper. The formula we derived assumes that the original rectangular sheet of paper is a rectangle.

Q: Can I use the formula to find the area of the resultant paper if the cut squares are not cut from the corners of the original rectangular sheet of paper?

A: No, the formula we derived assumes that the cut squares are cut from the corners of the original rectangular sheet of paper. If the cut squares are not cut from the corners of the original rectangular sheet of paper, you will need to use a different formula to find the area of the resultant paper.

Q: How do I find the area of the resultant paper if I don't know the length and width of the original rectangular sheet of paper?

A: If you don't know the length and width of the original rectangular sheet of paper, you can try to find it by measuring the shape of the resultant paper. Alternatively, you can use a different formula to find the area of the resultant paper.

Q: Can I use the formula to find the area of the resultant paper if the cut squares are not cut from a rectangular sheet of paper?

A: No, the formula we derived assumes that the cut squares are cut from a rectangular sheet of paper. If the cut squares are not cut from a rectangular sheet of paper, you will need to use a different formula to find the area of the resultant paper.

Q: How do I find the area of the resultant paper if I want to cut multiple squares from the corners of the original rectangular sheet of paper?

A: If you want to cut multiple squares from the corners of the original rectangular sheet of paper, you will need to use a different formula to find the area of the resultant paper. The formula we derived assumes that only one square is cut from each corner of the original rectangular sheet of paper.

Q: Can I use the formula to find the area of the resultant paper if the cut squares are not cut from the corners of the original rectangular sheet of paper, but from the edges?

A: No, the formula we derived assumes that the cut squares are cut from the corners of the original rectangular sheet of paper. If the cut squares are not cut from the corners of the original rectangular sheet of paper, but from the edges, you will need to use a different formula to find the area of the resultant paper.

Conclusion

In conclusion, the formula we derived can be used to find the area of the resultant paper if the cut squares are cut from the corners of a rectangular sheet of paper. However, if the cut squares are not cut from the corners of the original rectangular sheet of paper, or if the original rectangular sheet of paper is not a rectangle, you will need to use a different formula to find the area of the resultant paper.