How to Optimize the Areas & Perimeters of Rectangles

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  • 0:04 Area and Perimeter
  • 1:15 Maximum Area With…
  • 2:29 Minimum Perimeter With…
  • 3:21 Lesson Summary
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Lesson Transcript
Instructor: Betsy Chesnutt

Betsy teaches college physics, biology, and engineering and has a Ph.D. in Biomedical Engineering

In this lesson, learn about how to maximize the area of a rectangle when you know the perimeter. There is a simple trick that will help you get the biggest area every time, and this lesson will show you how to do it!

Area and Perimeter

John wants to plant a new garden and he has exactly 100 feet of fencing that he can use, so he knows that the perimeter of his garden must be 100 feet. The perimeter of any geometric shape (like John's new garden) is the length of all the sides of the shape added together.

John wants his garden to be rectangular, and he also wants to make the area, or total amount of space inside the fence, as large as possible. Since we know that he wants his garden to be a rectangle, the area can be found by multiplying the length of the rectangle times its width, or: A = L x W. You can see some examples of this appearing along with the formula below.

area and perimeter

As you can see, one rectangle with sides of 40 feet and 10 feet has an area of 400ft2., and the other rectangle with sides of 30 feet and 20 feet has an area of 600ft2.. Their perimeters are the same - both 100 feet overall - but their areas are different because of how their sides are arranged. So, to return to our opening example, it's clear that there are many different ways that he could arrange the fencing, and each would have a different area. How can he know which one will have the maximum area? We can help him with a little bit of math.

Maximum Area with Fixed Perimeter

If the garden is rectangular, it will have the largest possible area when the length equals the width. In order to have a perimeter of 100 feet, that means that each side needs to be 25 feet long. The area would then be 25ft x 25ft, or 625ft2.

area and perimeter of a square garden

This is true not just for John's garden, but for any rectangle. For a given perimeter, the area will be maximized when all the sides are the same length, which makes it actually a square. A square is still a rectangle, though!

So, if you know the perimeter, divide it by four to determine the length of each side. Then multiply the length times the width to get the area.

Let's look at one more example. Erin is building a rectangular fenced-in area for her dog, and she has 160 feet of fencing. What should the dimensions of her fence be if she wants to maximize the area her dog has to run and play, and what's the area enclosed by the fence?

You know that all the sides should be the same length, so to find the length of each side, divide 160 feet by 4. The fence should be 40 feet long on each side, which means that the total area enclosed by the fence will be (40ft) x (40ft), or 1600ft2.

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