Measuring area with unit squares
If everyone covers shapes with whatever unit they like, nobody’s counts agree. Which unit should everyone use, and how do you count it when the squares do not fit the shape exactly?
Where you are 15 in Area and perimeter
When you reach for it
In Area as covering the unit was free to choose: sticky notes, cards, paper triangles. That freedom is also the trouble with it. Eino says his drawing is 14 and Hilkka says hers is 30, and neither number means anything to anyone who has not seen their sticky notes. For a count of area to mean the same thing everywhere, everyone has to cover with the same unit.
The unit everyone agrees on is a square whose sides are one unit of length long: a square 1 centimeter on each side, or 1 inch, 1 foot or 1 meter. Its area is called one square centimeter (or square inch, square foot, square meter). This skill is measuring area with these squares. You reach for it whenever a shape is drawn on grid paper, laid out in square tiles, or small enough to trace onto a grid: count the squares that cover it, and when the squares do not fit the shape exactly, put the part squares together, or estimate.
The lesson takes three days. Part 1 sets up the square units and counts whole squares on a grid. Part 2 counts shapes made of whole squares and half squares. Part 3 estimates the area of a shape with a curved edge, where the squares can never fit exactly.
The procedure
Part 1: square units, and counting them on a grid
Why a square. Any unit that covers without gaps or overlaps would measure area, so why a square? Two reasons.
First, squares fit together edge to edge in rows with no gaps and no overlaps, and they keep doing so across a whole flat surface, however large. Round units cannot do this at all.
Second, and this is the deeper reason, a square matches the units of length. A square centimeter is a square whose side is exactly 1 centimeter. So along an edge 5 centimeters long, exactly 5 square centimeters fit in a row, side by side. The ruler and the area unit speak the same language: a length of 1 centimeter goes with an area of 1 square centimeter. A triangle also covers a surface without gaps, but no side of it lines up with a ruler in this neat way, so its count would not connect to the lengths of the shape. That match between the side of the square and the unit of length is what Area of rectangles and squares will use.
Naming the unit. The square unit takes its name from the length of its side.
- A square centimeter is a square 1 centimeter on each side, about the size of a fingernail.
- A square inch is a square 1 inch on each side, about the size of a postage stamp.
- A square foot is a square 1 foot on each side, the size of a big floor tile.
- A square meter is a square 1 meter on each side, big enough for a small child to lie curled up on.
An area is always given with its unit: “14 square centimeters”, never just “14”, and never “14 centimeters”. A centimeter is a length, the distance along a line; a square centimeter is a patch of flat space.
Counting on a grid. Centimeter grid paper is already covered by square centimeters, with no gaps and no overlaps. So a shape drawn along the grid lines is covered exactly by the grid squares inside it, and its area in square centimeters is the number of those squares. Count them in whatever groups are easy, such as row by row, and add. Marking each square as it is counted keeps any square from being counted twice or missed.
When the squares happen to sit in a rectangle of rows that are all the same length, the count can be done faster, by adding the rows or multiplying, as in Arrays. That shortcut is the next lesson. Here the shapes are not rectangles, and the count is the meaning.
Bigger square, smaller count. A square inch is bigger than a square centimeter (see Example 2), so a shape takes fewer square inches than square centimeters to cover. As in Area as covering, the unit changes the count but not the area.
Part 2: half squares
A shape on grid paper does not always follow the grid lines. Its edge may cut straight across a grid square from one corner to the opposite corner. That cut splits the square into two triangles. The two triangles are the same: turn one over and it fits exactly on the other. So each one is half a square unit.
Two such halves, put together, make one whole square unit. This is the fact from Area as covering that moving pieces, with no overlaps and none lost, keeps the area. So a shape made of whole squares and half squares is measured by counting both kinds:
- the whole squares count 1 square unit each;
- the half squares are paired off, and each pair counts as 1 square unit;
- if one half is left over, it adds half a square unit.
A square cut down the middle, edge to edge, also makes two halves: two thin rectangles, each half a square unit. A half made one way and a half made the other way have the same area, because each one is exactly half of the same square, and two of either kind fill it.
A cut that runs from a corner to some other point does not make halves. Only call a piece half a square when two copies of it fill the square.
Part 3: estimating a shape with a curved edge
A leaf, a puddle or a round coaster has a curved edge. Wherever the curve crosses a grid square, it cuts off a piece of some odd size, and no amount of pairing makes these pieces into exact wholes. The area can still be found closely, by estimating.
Two counts trap the area between them.
- Counting only the full squares, the ones wholly inside the shape, gives too little, since it leaves out every part square along the edge.
- Counting every square the shape touches gives too much, since it counts each part square as a whole one.
The true area is more than the first count and less than the second.
A good estimate sits between them. To get one, we need a rule for the part squares, and the rule is a choice, an agreement that keeps everyone’s counts alike, not a fact about the shape. The usual choice, used in this lesson, goes like this. Look at each part square along the edge. If the shape covers more than half of it, count it as a whole square. If the shape covers less than half, leave it out. Then add the full squares and the more-than-half squares. Other fair rules exist (counting every part square as a half, for instance), and they give estimates that differ a little, but all of them land between the two bounds.
This works because the errors pull in opposite directions. A square that is mostly covered, counted as a whole, adds a little too much. A square that is barely covered, left out, leaves out a little. All along the curve the too-much bits and the too-little bits roughly cancel, so the total lands close to the true area. It is still an estimate, and the answer says so: “about 23 square centimeters”.
A finer grid gives a closer estimate. With smaller squares, each part square is smaller, so each error is smaller.
Worked examples
Part 1: a shape on centimeter grid paper
Joona draws the shaded shape on centimeter grid paper, along the grid lines. What is its area?
Each grid square is one square centimeter, and the shape is covered by the grid squares inside it, with no gaps and no overlaps. So its area is the number of those squares, in square centimeters.
Count row by row. The top row has 4 squares. The middle row has 5. The bottom row has 5, shifted one square to the right. The rows do not overlap, so their counts add:
The area is 14 square centimeters.
Check: count the columns instead, from left to right: 2, 3, 3, 3, 2, 1, and 2 + 3 + 3 + 3 + 2 + 1 = 14.
Part 1: a square inch and a square centimeter
Hilkka draws a square 1 inch on each side on centimeter grid paper, with one corner on a corner of the grid. Which is bigger, a square inch or a square centimeter? About how many square centimeters fit in one square inch?
An inch is longer than a centimeter: one inch on a ruler is a little more than 2 and a half centimeters. So the side of the square inch runs past 2 grid lines and stops short of the third.
Inside the square inch lie 2 rows of 2 whole square centimeters, which is 2 + 2 = 4. Along the right edge and the bottom edge are thin strips, and in the corner a tiny piece, all inside the square inch. So the square inch is more than 4 square centimeters.
The square inch sits inside a block of 3 rows of 3 grid squares, which is 3 \times 3 = 9 square centimeters, and does not fill it. So it is less than 9 square centimeters. The two strips together are about 2 more square centimeters, so the square inch is about 6 square centimeters.
A square inch is bigger than a square centimeter: it holds between 4 and 9 of them, about 6. A shape measured in square inches gets a smaller count than the same shape in square centimeters.
Part 1: a tiled bathroom floor
Eino’s bathroom floor is covered by square tiles, each 1 foot on a side, with no gaps. The three rows by the door have 6 tiles each. The two rows beside the bathtub have 4 tiles each. What is the area of the floor? Why is the square foot a sensible unit here, and not the square inch?
SolutionEach tile is a square 1 foot on each side, so each tile is 1 square foot, and the tiles cover the floor exactly. The area is the number of tiles. Add the rows:
The floor is 26 square feet.
A square inch is a much smaller square, so the same floor would take a huge number of square inches, far too many to count. A big surface is measured in a big unit (square feet or square meters) and a small one, like a drawing, in a small unit (square centimeters or square inches), so that the count stays a size people can picture.
Part 2: a crown of whole and half squares
Hilkka draws a crown on centimeter grid paper. It is 2 rows of 5 whole squares, with two pointed peaks on top. Each peak is made of 2 half squares, cut corner to corner. What is the area of the crown?
The whole squares are 5 + 5 = 10 square centimeters.
There are 2 peaks of 2 half squares each, so 2 + 2 = 4 half squares. Each half is half a square centimeter, since the corner-to-corner cut splits a grid square into two matching triangles. Pair them: 4 halves make 2 pairs, and each pair fills one square centimeter. So the halves add 2 square centimeters. (Each peak is itself one pair: slide one of its halves next to the other and the two fill a grid square.)
The crown is 12 square centimeters.
Part 2: a triangle on the grid
Joona draws a triangle on centimeter grid paper. Its two short sides run along grid lines, each 4 centimeters long, and its long side runs from corner to corner through 4 grid squares. What is its area?
Sort the grid squares inside the triangle into two kinds.
The long side passes through 4 squares, and in each it runs exactly from one corner to the opposite corner. So it cuts each of those squares in half, and the triangle holds one half of each: 4 half squares, which pair off into 2 square centimeters.
Below the long side, the whole squares sit in a staircase, in rows of 1, 2 and 3:
Together:
The triangle is 8 square centimeters.
Check: two copies of this triangle fit together into a square 4 squares on each side, which holds 4 rows of 4, or 16 square centimeters. Half of 16 is 8.
Part 3: a traced leaf
Eino traces a leaf onto centimeter grid paper. He shades the squares that lie wholly inside the outline and marks with a dot each square that is more than half inside. Estimate the area of the leaf.
The edge of the leaf is curved, so the squares along it are cut into pieces of every size, and an exact count is not possible. Estimate instead.
Count the full squares: there are 13, shaded in the figure.
Count the squares more than half covered, the dotted ones: there are 10. Each is counted as a whole square. The other squares along the edge are less than half covered, and are left out.
The leaf is about 23 square centimeters. The extra bits counted in the dotted squares and the slivers left out of the others roughly cancel, so the estimate is close. It cannot be less than the 13 full squares.
Part 3: trapping a coaster’s area
Hilkka traces around a round coaster on centimeter grid paper. The circle lies wholly over 13 grid squares, more than half over 8 others, and touches 37 squares in all. Between which two numbers must the area lie? What is a good estimate?
The 13 full squares are all inside the circle, and there is more circle beyond them, in the part squares along the edge. So the area is more than 13 square centimeters.
The 37 touched squares hold the whole circle, but the ones along the edge, beyond the 13 full squares, are only partly covered. So the area is less than 37 square centimeters.
The area lies between 13 and 37 square centimeters. For the estimate, count the full squares and the more-than-half squares:
The coaster is about 21 square centimeters, which sits between the two bounds, as it must. The bounds are far apart, 37 - 13 = 24, because a small circle has many squares along its edge; the estimate is much closer to the truth than either bound.
Boundaries
A grid count measures area only if the squares are all the same size. Grid paper is printed that way; a hand-drawn grid with uneven squares does not measure in any one unit.
The unit must be named. “14” alone is not an area. “14 square centimeters” and “14 square inches” are different areas, since the units are different sizes.
A square centimeter is not a centimeter. One measures flat space, the other length along a line. A length and an area are never added to each other.
Only a cut that makes two matching pieces makes halves. A corner-to-corner cut and a cut through the middle from edge to edge both do. Other cuts make pieces of other sizes, and those cannot be paired as halves.
Counting is exact only for shapes made of whole and half squares. A curved edge, or a slanted edge that does not pass through corners, leaves part squares that do not pair into wholes. Then the count gives an estimate, which should be called “about”, and which lies between the full-square count and the touched-square count.
Where this goes next
When the squares covering a shape sit in a rectangle, all the rows have the same number of squares. Counting them then becomes adding equal rows, or multiplying the number of rows by the number in each row. Because a square unit’s side is one unit of length, those two numbers are just the rectangle’s side lengths. That is Area of rectangles and squares, and cutting a rectangle into its rows and columns of squares is Splitting a rectangle into rows and columns of squares.
The square unit also makes clear that Perimeter and area measure different things: the perimeter counts unit lengths around the edge, the area counts unit squares inside, as Perimeter versus area explores. Changing a count from one square unit to another, such as square feet into square inches, is Converting square and cubic units.
The half squares of Part 2 grow into whole lessons. Example 5 found a triangle to be half of a square, and Area of triangles shows that every triangle is half of a related rectangle or parallelogram. The estimate of Part 3, with finer and finer grids, is how the area of a circle is first pinned down, as in Why the circle area formula works.
Exercises
Part 1: square units and counting on a grid
The figure shows shapes A, B and C on centimeter grid paper, for Exercises 1 to 3.
- Find the area of shape A.
- Find the area of shape B.
- Find the area of shape C, the ring. Do not count the hole.
- Which unit, square centimeters, square inches, square feet or square meters, would you choose for each of these: a postage stamp, a page of a notebook, a classroom floor, a soccer field? Say why.
- Hilkka’s porch floor is covered by square tiles, each 1 meter on a side, with no gaps. It has rows of 5, 5 and 3 tiles. What is the area of the porch floor?
- Eino makes a mosaic from square tiles 1 centimeter on a side, laid edge to edge. It has 3 rows of 6 red tiles and, below them, 2 rows of 6 blue tiles. What is the area of the mosaic?
- Joona draws a shape with an area of 9 square centimeters. Hilkka draws a shape with an area of 9 square inches. Whose shape is bigger? Explain.
Part 2: half squares
The figure shows shapes D, E and F on centimeter grid paper, for Exercises 8 to 10.
- Find the area of shape D, the arrow.
- Find the area of shape E, the slanted shape.
- Find the area of shape F, the diamond.
- Joona’s quilt square is sewn from patches. It has 12 whole square patches, each 1 inch on a side, and 6 triangle patches, each half of such a square, cut corner to corner. What is the area of the quilt square?
- A shape on centimeter grid paper is made of 7 whole squares and 5 half squares. What is its area?
- Hilkka’s shape has an area of 15 square centimeters. It is made of 11 whole squares and some half squares. How many half squares are there?
Part 3: estimating
- Joona chalks a grid of 1-foot squares on the playground around a puddle (see the figure). Count the squares wholly inside the puddle, and the squares more than half inside it. Estimate the area of the puddle.
- Hilkka traces a flat stone on centimeter grid paper. She counts 9 full squares, 5 squares more than half covered, and 6 squares less than half covered. Estimate the area of the stone. Between which two numbers must the true area lie?
- Eino traces his shoe on centimeter grid paper. He counts 78 full squares, 13 squares more than half covered, and 11 squares less than half covered. Estimate the area of the shoe’s outline.
- A shape drawn on centimeter grid paper touches 30 squares in all, and 18 of them are full. Between which two numbers must its area lie? If 7 of the other squares are more than half covered, what is a good estimate?
- Eino traces the same leaf on centimeter grid paper and on inch grid paper. Which grid gives the closer estimate, and why?
Reasoning
- A paper triangle can also cover a surface with no gaps and no overlaps. Why is the standard unit of area a square, with sides one unit of length, and not a triangle?
- Why is an area written as “12 square centimeters” and not “12 centimeters”?
- Joona cuts one square centimeter corner to corner. Hilkka cuts another down the middle, edge to edge. Is one of Joona’s pieces the same area as one of Hilkka’s? Explain.
- When estimating, why is it fair to count a square that is more than half covered as a whole square, and to leave out one that is less than half covered? Is the estimate exact?
Answers
- 8 square centimeters. The top row has 5 squares and the column below it has 3 more: 5 + 3 = 8.
- 10 square centimeters. The rows add: 1 + 2 + 3 + 4 = 10.
- 12 square centimeters. The top and bottom rows have 4 squares each, and the two middle rows have 2 each, one at each side of the hole: 4 + 4 + 2 + 2 = 12.
- A postage stamp: square centimeters (or square inches), since it is tiny. A notebook page: square inches or square centimeters. A classroom floor: square feet or square meters. A soccer field: square meters, since it is huge. A small unit for a small surface and a big unit for a big one keeps the count a sensible size.
- 13 square meters. Each tile is 1 square meter, and 5 + 5 + 3 = 13.
- 30 square centimeters. The red tiles are 3 \times 6 = 18, the blue tiles are 2 \times 6 = 12, and the two parts do not overlap, so 18 + 12 = 30.
- Hilkka’s. Both shapes are 9 squares, but a square inch is bigger than a square centimeter (about 6 square centimeters), so 9 square inches cover more space.
- 7 square centimeters. The block is 3 + 3 = 6 whole squares, and the point is 2 half squares, which make 1 more: 6 + 1 = 7.
- 10 square centimeters. Each row has 4 whole squares and 2 halves, so 8 whole squares and 4 halves in all. The halves make 2, and 8 + 2 = 10.
- 8 square centimeters. The middle holds 4 whole squares, and each side of the diamond cuts 2 squares in half, so there are 8 half squares, which make 4. Then 4 + 4 = 8.
- 15 square inches. The 6 triangles pair into 3 whole squares, and 12 + 3 = 15.
- \displaystyle 9\frac{1}{2} square centimeters. Four of the halves pair into 2 squares, with one half left over: 7 + 2 = 9, and half a square more.
- 8 half squares. The halves must make 15 - 11 = 4 square centimeters, and each square centimeter is 2 halves: 4 \times 2 = 8.
- About 22 square feet. The puddle lies wholly over 13 squares and more than half over 9 others, and 13 + 9 = 22. (It touches 33 squares in all, a few of them only by thin slivers at the ends, so the area is between 13 and 33 square feet.)
- About 14 square centimeters, since 9 + 5 = 14. The true area is more than the 9 full squares and less than all 9 + 5 + 6 = 20 touched squares.
- About 91 square centimeters: 78 + 13 = 91. The squares less than half covered are left out. (The true area is more than the 78 full squares and less than all 78 + 13 + 11 = 102 touched squares.)
- More than 18 and less than 30 square centimeters. The estimate is 18 + 7 = 25 square centimeters.
- The centimeter grid. Its squares are smaller, so each part square along the edge is smaller, and counting it as whole or leaving it out makes a smaller error.
- A square’s side is one unit of length, so square units line up along the sides of a shape: a 5-centimeter edge holds exactly 5 square centimeters in a row. That ties the count of squares to the lengths measured with a ruler. A triangle covers too, but its sides do not line up with the grid of lengths in rows and columns this way.
- A centimeter is a length, measured along a line. The area counts squares 1 centimeter on a side, which are patches of flat space, so the unit is the square centimeter.
- Yes. Each piece is exactly half of a square centimeter, because two copies of it fill the square. So every one of the four pieces is half a square centimeter, whatever its shape.
- The rule is a choice everyone agrees on, and a fair one. Counting a mostly covered square as whole adds a little too much; leaving out a barely covered one leaves out a little. Along the curved edge these errors pull in opposite directions and roughly cancel, so the total is close. It is not exact: it is an estimate, between the full-square count and the touched-square count.