To find the largest possible area of a rectangle for a given perimeter, you should shape the rectangle into a square.
For a fixed perimeter, a rectangle achieves its maximum area when all four of its sides are equal in length, making it a square. You can find this maximum area by first determining the side length of the square and then calculating its area.
Steps to Maximize Area for a Given Perimeter
Here’s how to find the dimensions that yield the largest area and calculate that area:
- Determine the Side Length: Since a square has four equal sides, take the given perimeter and divide it by four. This gives you the length of each side of the square.
- Formula:
Side Length = Perimeter / 4
- Formula:
- Calculate the Maximum Area: Once you have the side length (which is both the length and the width of the square), calculate the area by multiplying the length times the width.
- Formula:
Maximum Area = Side Length × Side Length(orSide Length²)
- Formula:
This principle is derived from the fact that, for a constant sum (the perimeter), the product (the area) is maximized when the numbers being multiplied are as close to each other as possible. When length equals width, they are perfectly equal, resulting in a square.
Example: Maximizing Area with a Perimeter of 20 units
Let's say you have a perimeter of 20 units and want to find the rectangle with the largest possible area.
- Given Perimeter: 20 units
- Step 1: Find Side Length: Divide the perimeter by 4.
Side Length = 20 / 4 = 5 units
- Step 2: Calculate Maximum Area: Multiply the side length by itself.
Maximum Area = 5 units × 5 units = 25 square units
The rectangle with a perimeter of 20 units that has the largest area is a square with sides of 5 units each. Its area is 25 square units.
Comparing Areas for a Fixed Perimeter
Consider other rectangles with a perimeter of 20 units:
| Length | Width | Perimeter (2L + 2W) | Area (L × W) |
|---|---|---|---|
| 5 | 5 | 2(5) + 2(5) = 20 | 5 × 5 = 25 |
| 6 | 4 | 2(6) + 2(4) = 20 | 6 × 4 = 24 |
| 7 | 3 | 2(7) + 2(3) = 20 | 7 × 3 = 21 |
| 8 | 2 | 2(8) + 2(2) = 20 | 8 × 2 = 16 |
| 9 | 1 | 2(9) + 2(1) = 20 | 9 × 1 = 9 |
As the table shows, for a perimeter of 20, the square (Length=5, Width=5) indeed yields the largest area.
Understanding Rectangle Area
The fundamental way to find the area of any rectangle is by multiplying its length by its width. However, the question of finding the largest area implies a constraint, and as shown, the constraint of a given perimeter leads to the square as the shape that maximizes this product (Area = Length × Width).
For more information on geometric shapes and their properties, you can explore resources like Khan Academy Geometry or other educational websites.
