Question: What is the largest the area of a rectangle can be if its perimeter is 20?

Discover the answer with this applet. Find the answer and a more challenging question below.

Also use this applet to examine the relation between length and width when the area, perimeter, or both are held constant. Use the line of constraint to draw a line on which the upper right corner of your rectangle will remain. Also use the trace with the hold area and perimeter buttons to see the line or curve created by this corner.

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Use a 14 by 18 Graph

Use a 70 by 90 Graph

Answer: The largest area of any rectangle with a given perimeter will occur when a square is formed. With a perimeter of 20, a square will form an area of 25. This is the largest area a rectangle can have with a perimeter of 20. Go back to the applet if you have not yet discovered this.

More Challenging Question: What is the largest the perimeter of a rectangle can be if its area is 10? Look below for a hint, and under that for the answer.

Hint: You must think about the grid extending farther than this applet shows. (Actually the grid can extend infinitely). Try first using the applet and looking at the same question with a rectangle with an area of 1. Use the hold area button.

Answer: The perimeter can be infinite. Think of the width of the rectangle as very long, say 10,000,000. As long as the length is a very small fraction, we can achieve an area of 10. In this case, the length would have to be 1/1,000,000. The width can become infinitely large as the length approaches 0 and vice versa. So we can see the perimeter can be infinite.