Question

What are the side lengths of a rectangle if the area = 40 in and the perimeter= 26 in

Answer 1

Let's figure this out by first defining area and perimeter.  Perimeter is the sum of all the side lengths of a figure.  In a rectangle, the perimeter is equal to the length plus the length plus the width plus the width: l+l+w+w, or just 2l+2w.  This is because in a rectangle, opposite sides have the same length.  Area is how much space is inside the rectangle: the length times the width, or l*w.

So, we need to figure out a value for l and a value for w so that 2l+2w=26, and l*w=40.  Let's start with l.  If l=1, then the equation is (2*1)+2w=26.  2*1=2, so 2+2w=26.  Subtract 2 from both sides: 2w=24.  Divide by 2 on both sides: w=12.  So let's see if this works, where l=1 and w=12.  1*12=12.  12 is not equal to 40, so that doesn't work.

What about l=2?  We can solve this in a similar way.  (2*2)+2w=26.  4+2w=26.  2w=22.  w=11.  Does this work?  l*w=2*11=22.  22 does not equal 40.

For l=3: (2*3)+2w=26.  6+2w=26.  2w=20.  w=10.  3*10=30, not 40.  This isn't right either.  However, do you see a pattern?  Every time you add one to l, w goes down by 1.  This is because perimeter is a sum!

So if l=4, then w=9.  4*9=36, which does not equal 40.  But we're getting closer!

If l=5, then w=8.  5*8=40!!  So, our length is 5 inches, and the width is 8 inches.  There's your answer!

Answer: 5 inches and 8 inches