Expanding a Square Garden: Calculating New Areas and Understanding Shape Changes

When a gardener decides to expand the area of a square garden by increasing its side length, the process involves understanding the initial area and calculating the new area based on the given modifications. This article will explore the mathematical calculations involved and the potential changes in the shape of the garden.

Understanding the Original Area

Let's denote the original side length of the square garden as s. The original area of the garden, denoted as ( text{Original Area} ), is calculated as:

(text{Original Area} s^2)

Increasing the Side Length

Suppose the side length of the garden is increased by 25%. The new side length is calculated as:

(text{New Side Length} s 0.25s 1.25s)

Calculating the New Area

The new area of the garden is then calculated as:

(text{New Area} (1.25s)^2 1.5625s^2)

In terms of the original area, the new area can be expressed as:

(text{New Area} 1.5625 times text{Original Area})

This indicates that the new area is 156.25% of the original area. For example, if the original area is 100 square units, the new area would be:

(text{New Area} 1.5625 times 100 156.25 text{ square units})

Exploring Shape Changes

It is essential to consider the potential changes in the shape of the garden when increasing the side length of only one side. For example, if a side of 8 meters is extended by 4 meters, the garden will no longer be a square but a rectangle. The new dimensions would be 8 meters by 12 meters, making the new area:

(text{New Area} 8 times 12 96 text{ square meters})

Forming a Trapezium

When only one side of the square garden is increased, the garden will no longer be a square but a trapezium. The extension will result in the addition of a right-angled triangle to one side of the square. The area of this triangle can be calculated as:

(text{Area of Triangle} frac{1}{2} times 4 times 8 16 text{ square meters})

The total area of the increased garden would then be:

(text{Total New Area} 8 times 8 16 80 text{ square meters})

Mathematical Representation

Mathematically, if the original side length of the square garden is (s) and it is increased by 25%, the new area can be represented as:

(text{New Area} 1.25^2s^2 1.5625s^2)

This represents a 56.25% increase in the original area.

In conclusion, expanding a square garden by increasing one side length by 25% results in a new area that is 156.25% of the original area. This can be applied to practical scenarios in gardening, such as increasing the dimensions of a garden bed to accommodate more plants, vegetables, or flowers. Understanding these calculations helps gardeners plan and design their gardens effectively.