Question: The perimeter of a rectangle is 72. What is the largest possible area of the rectangle?

Sun, 03 May 2026 02:30:51 +0000

**The perimeter of a rectangle is 72. What is the largest possible area of the rectangle?**
This everyday geometry question isn’t just academic—it’s popping up in home projects, app design, and classroom learning across the U.S. Curious about how to get the most space from a fixed boundary? Understanding the perfect rectangle under these constraints reveals surprising insights into efficient design and mathematical optimization. **Why This Question Is Gaining Attention in the US**
In an era focused on smarter spatial planning—from minimalist homes to efficient packaging—this problem resonates with real-world challenges. Mobile users searching for practical math applications find clarity here. As stereotypes about math stay relevant, the perimeter-and-area puzzle serves as a gateway to seeing geometry’s value beyond schoolbooks. It’s a subtle but powerful example of how structured thinking leads to smarter decisions. **How the Perimeter-Area Equation Works**
Perimeter measures the total boundary length around a shape. For a rectangle, P = 2(length + width), and we know P = 72. To find maximum area (A = length × width), express one variable in terms of the other: width = (72 – 2×length)/2 = 36 – length. Substituting gives A = length × (36 – length). This creates a quadratic equation: A = –length² + 36length. The maximum of this curve occurs at its vertex—exactly halfway between the two sides. Solving gives length = 18, so both sides are 18: the optimal shape is a square. This gives the largest area of 324 square units. **Common Questions People Ask About This Problem**
- *Why isn’t a long, skinny rectangle bigger?* Because the perimeter limits how far wide or tall the shape can grow—spread evenly yields more space.
- *Does this apply to real construction or design?* Absolutely—from open living spaces to shipping containers, knowing how to maximize area under fixed perimeters ensures efficiency and cost savings.
- *What if dimensions aren’t whole numbers?* The math still holds—incremental adjustments within the perimeter constraint preserve maximum area, always settling at values closest to equal sides.