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!
a dodecagon has 12 sides, now, if we take a circular 360° and divide into 12 even parts, we get 360/12 = 30, 5 of those angles added up will just be 150°, Check the picture below.
