Folded paper

A rectangular strip of paper measures 26 centimetres by 8 centimetres. One end of the strip is folded over. The folded piece is 6 centimetres long. It lies flat on top of the rest of the strip. Find the area of the shape that can now be seen from above. Watch the end fold over. Now look down. What left the view is the flap, once. The paper underneath it is still paper. The dashed outline is the place the flap came from. The flap is the same size as that place. So write the rule. Area seen from above is the area of the strip minus the flap. Start with the whole strip. 26 times 8 is two hundred and eight square centimetres. The flap is 6 by 8. That is 48 square centimetres. Two hundred and eight minus 48 is one hundred and sixty. The area seen from above is one hundred and sixty square centimetres.

Now there are two folds. A rectangular strip of paper measures 30 centimetres by 9 centimetres. Both ends are folded inwards. Each fold is 7 centimetres. The two flaps lie flat on the strip. They do not meet. Find the area of the shape that can now be seen from above. Area seen from above is the area of the strip minus both flaps. 30 times 9 is two hundred and seventy square centimetres. That is the whole strip. One flap is 7 by 9. That is 63 square centimetres. There are two folds. Area of two flaps is 63 times 2. That is one hundred and twenty six square centimetres. Two hundred and seventy minus one hundred and twenty six is one hundred and forty four. The area seen from above is one hundred and forty four square centimetres.

Your turn. A rectangular sheet of paper A B C D measures 20 centimetres by 12 centimetres. It is folded once. The corner A lands on the point P inside the sheet. The crease X Y has X on A B and Y on A D. The lengths A X and A Y are equal. One shaded triangle is the gap left at the corner. The other shaded triangle is the folded flap lying on the sheet. The gap and the flap have a total area of 64 square centimetres. Find the length of A X. Try it first. Pause if you like. The corner moved across the crease. The dashed triangle is the place it came from. The flap is the same size as that place. So the gap and the flap are identical. Area of one shaded triangle is 64 divided by 2. That is 32 square centimetres. The flap has two equal short sides. So half of A X times A X is 32. A X times A X is A X squared. That is 64. 8 times 8 is 64, so the square root of 64 is 8. A X is 8 centimetres. The dashed triangle is where the flap came from. The shaded triangle is where it landed. They are the same piece of paper. A fold is one sheet, in two layers.