Area on squared paper

How do you find the area of a curvy shape, like a leaf? Use squared paper (or graph paper), where each small square is 1 unit by 1 unit, that is 1 square unit. Trace the shape onto transparent paper, place it on the squared paper and count the squares inside it.

Most shapes do not fit the squares neatly, so the book gives four rules:

• 1: a full small square counts as 1 sq unit.
• 2: a part of a square that is less than half is ignored.
• 3: if more than half of a square is inside the shape, count it as 1 sq unit.
• 4: if exactly half a square is inside, count it as 1/2 sq unit.

It helps to tick each square as you count it, so none is counted twice.

For example, suppose a leaf shape covers 8 full squares, 3 squares that are more than half inside, 2 squares that are exactly half inside, and 4 squares that are only a little bit inside (less than half). Its area is about 8 + 3 + 1/2 + 1/2 + 0 = 12 sq units.

Why squares? A girl in the book asks why we can't use circles instead. Circles can't be packed tightly: they always leave gaps in between. The book packs the same rectangle with circles in two different ways, and gets 42 circles one way and 44 circles the other. With gaps, and a different answer each time, circles don't give an accurate area. Squares fit together with no gaps and no overlaps, so they make a good unit for measuring area.