Grade 3 · 3.MD.C.5 · 3.MD.C.5B

Area as covering

A placemat, a picnic blanket and a puzzle piece each take up some flat space. How do you say how much, and how can you tell which of two shapes takes up more?

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Iterating a single unitGrade 1

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Measuring area with unit squaresGrade 3 Perimeter versus areaGrade 3

Where you are 15 in Area and perimeter

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Area and perimeter15

Area as covering

Measuring area with unit squares Splitting a rectangle into rows and columns of squares Area of rectangles and squares Perimeter ▾ Show all 15 (10 more)

Why this exists

Tomoya wants to know how big his placemat is. A ruler tells him how long it is, and how wide, but neither number on its own says how much table it covers. A long thin placemat and a short wide one can have the same length and still cover very different amounts of table.

What Tomoya is after is a different kind of size: how much flat space the placemat takes up. That kind of size is called area.

Length already has a way to be measured. To measure a length you pick a unit, such as a paper clip, and lay copies of it end to end along the thing, counting as you go (see Iterating a single unit). Area is measured the same way, one step up. You pick a small flat shape as the unit, lay copies of it over the whole surface so that they cover it, and count the copies. Tomoya has a pad of square sticky notes. He covers the placemat with them, and the count, 12, is its area: 12 sticky notes.

This one idea answers questions that come up all the time. How much paper covers a desk? Which of two blankets gives more room to sit? If a shape is cut up and its pieces are moved, is there more of it, less, or the same? The lesson takes these in three parts, one for each day: covering and counting, choosing a unit and comparing, and cutting and rearranging.

Definition

The area of a flat shape is the amount of flat space inside it.

To measure an area, choose a unit: one small flat shape, such as a sticky note, a paper triangle or a paper card. Then cover the shape with copies of the unit so that

there are no gaps: every bit of the shape is under some copy, and
there are no overlaps: no copy lies on top of another, and none hangs over the edge.

The number of copies is the area, in that unit. A placemat covered exactly by 12 sticky notes has an area of 12 sticky notes.

The unit is named with the number, the way a length is “7 paper clips” and not just “7”. Every copy of the unit must be the same size and shape. A cover may turn or flip some copies, as long as each one is a copy of the same shape.

Intuition

The lesson runs over three days. Part 1 covers shapes and counts the copies. Part 2 asks what changes when the unit changes, and how two shapes are compared. Part 3 cuts shapes into pieces and moves the pieces.

Part 1: covering and counting

Here is Tomoya’s placemat, covered with sticky notes.

Figure 1Twelve sticky notes cover the placemat exactly, so its area is 12 sticky notes.

The notes touch edge to edge. None of the placemat shows between them, and none lies on top of another. So the count of notes is the area: 12 sticky notes.

The notes happen to sit in rows: 3 rows with 4 notes in each row. That is an array, so Tomoya does not have to count one by one; he can add 4 + 4 + 4 or find 3 \times 4, as in Arrays. But the array is a shortcut for counting, not the meaning of area. The meaning is the cover. A shape whose cover does not sit in neat rows still has an area, and you find it by counting the copies in whatever groups are handy.

The two conditions in the definition are what make the count honest. The figure below shows the same rectangle three times.

Figure 2Only the cover on the right measures the rectangle. Gaps leave space uncounted; an overlap counts some space twice.

On the left the notes are too small to meet, and the rectangle shows between them. Some of its space is under no note, so a count of 6 notes would miss that space. In the middle a seventh note lies across four others. The space under it is now counted twice, so a count of 7 is too big for the rectangle. Only the cover on the right, edge to edge with nothing doubled, gives the area: 6 sticky notes.

One more fact is clear from covering. If two shapes are set side by side without overlapping, the cover of the new, bigger shape is just the two covers together. So its area is the two areas added. A desk covered by 2 rows of notes, pushed against a desk covered by 3 rows of the same length, makes a desk covered by 5 rows; this is the picture behind the The distributive property.

Part 2: the unit, and comparing two shapes

Nothing in the definition says the unit has to be a sticky note. Avner cuts cards from stiff paper, each one made exactly as big as two sticky notes side by side, and covers the same placemat with them. Batsheva cuts each sticky note corner to corner into two triangles, and covers the placemat with those.

Figure 3One placemat, three counts. It is 12 sticky notes, 6 cards or 24 triangles, depending on the unit.

The placemat has not changed, but the counts have: 12 sticky notes, 6 cards, 24 triangles. All three are correct. Each tells the same area in a different unit.

The counts move in a pattern. A card is two notes, so it takes half as many cards as notes. A triangle is half a note, so it takes twice as many triangles as notes. A bigger unit gives a smaller count, and a smaller unit gives a bigger count. This is the same thing that happens with length: a desk is fewer hand spans long than paper clips long.

This is also why a count alone, with no unit, says nothing. “My blanket is 10” and “my blanket is 20” cannot be compared until we know 10 of what and 20 of what. To compare two areas by counting, measure both in the same unit, or change one count into the other unit first.

Comparing by covering also shows that area is not the same as length around. The two picnic blankets below are made of the same square patches, and the same length of trim fits around the edge of each one.

Figure 4The same length of trim goes around each blanket, but blanket B covers more ground.

Yet blanket B covers more ground. Blanket A is covered by 2 rows of 6 patches, which is 12 patches. Blanket B is covered by 4 rows of 4 patches, which is 16 patches. The distance around a shape, its Perimeter, is a length; the area is the space inside. Two shapes can share one and differ in the other, as Perimeter versus area explores.

Part 3: cutting and moving the pieces

Sometimes two shapes can be compared without any unit at all. Lay one on top of the other: if the first fits inside the second with room left over, the second has the larger area. If they match exactly, the areas are equal.

When neither fits inside the other, cutting helps. Avner has a long paper strip, covered by 8 sticky notes in a single row, and a mat covered by 2 rows of 4 notes. Which takes up more space?

Figure 5Cut the strip, slide one half, and it fits the mat exactly. Nothing was added and nothing was lost.

He cuts the strip in half and slides the right half under the left half. The two halves now fit the mat exactly, so the strip and the mat have the same area. Cutting did not make any paper and did not throw any away; moving the halves did not stretch or squash them. The amount of flat space stayed the same, and only the shape changed.

This is true of any shape cut into any pieces. Cutting a shape into pieces and moving the pieces, without overlapping them and without losing any, does not change the area. A tangram set shows it well. Its seven flat pieces, lettered here from A to G, fit together into a square. Batsheva slides big triangle A off the top edge and fits it under the bottom edge.

Figure 6The square and the arrow are made of the same seven pieces, so they have the same area.

The new shape looks like an arrow and nothing like a square. But it is the same seven pieces, so it covers the same amount of table. Any shape built from all seven pieces, a cat, a boat or a house, has the area of the square.

The tangram also shows that a unit can be any shape that covers. Take the small triangle, D or E, as the unit. Each of the other pieces can be cut into copies of it, so every piece, and every shape the set makes, has an area that is a whole number of small triangles. Example 7 finds them.

Results

Area is a count of units. If copies of a unit cover a flat shape with no gaps and no overlaps, the number of copies is the shape’s area in that unit.

Same unit, fair comparison. Two shapes measured in the same unit are compared by comparing their counts: the larger count is the larger area.

The unit changes the count, not the area. Measured in a bigger unit, the same shape takes fewer copies. If one big unit is exactly made of 2 small ones, the count in small units is 2 times the count in big units, and the same pattern holds for 3, 4 or any number.

Joining adds areas. Two shapes set together without overlapping make a shape whose area is the sum of their areas.

Moving pieces keeps the area. Cutting a shape into pieces and putting the pieces together in a new way, with no overlaps and no piece lost, gives a shape with the same area.

Why it is true

Each result comes from looking at the cover.

Same unit, fair comparison. Suppose one shape is covered by 14 notes and another by 12 notes of the same size. The cover of the first could be matched with the cover of the second, note for note, and 2 notes of the first would be left over. Those 2 notes cover space that the second shape does not have, so the first shape has more space inside it.

The unit changes the count. Suppose each card is exactly two notes. Cover a shape with cards, then cut every card down the middle into its two notes. Nothing moved, so the shape is now covered by notes with no gaps and no overlaps, and there are 2 notes for every card. The area is the same; only the way of counting it changed.

Joining adds areas. Put the two covers side by side along with the shapes. Since the shapes do not overlap, neither do their covers, and together the covers leave no gaps in the joined shape. So the joined shape is covered by all the copies from both covers, and its count is the two counts added.

Moving pieces keeps the area. Cover the shape before it is cut. Each piece takes its part of the cover with it. When the pieces are put together without overlapping, the parts of the cover do not overlap either, and since no piece is lost, every copy of the unit is still there. So the new shape is covered by the same number of copies as the old one. (If a cut runs through a copy of the unit, cut that copy too and count its parts as one; the parts still fill the same space as one copy.)

Worked examples

Example 1

Part 1: a placemat of sticky notes

Tomoya covers his placemat with square sticky notes. They fit with no gaps and no overlaps, in 3 rows of 4 notes. What is the area of the placemat?

Solution

The notes cover the placemat exactly, so the area is the number of notes, measured in sticky notes. The notes form an array of 3 rows, each a group of 4, so the count is 4 + 4 + 4 = 12, or, as a multiplication, 3 \times 4 = 12.

The area of the placemat is 12 sticky notes.

Example 2

Part 1: a desk top shaped like an L

Tomoya’s desk top is shaped like a capital L. He covers it with sticky notes, with no gaps and no overlaps. The top two rows have 5 notes each. Under them, at the left end, are two more rows of 2 notes each. What is the area of the desk top?

Solution

The cover is not one array, but the count is still the area. Count it in parts that are easy to see. The top two rows hold 5 + 5 = 10 notes. The two short rows hold 2 + 2 = 4 notes. The two parts do not overlap, so joining them adds their counts: 10 + 4 = 14.

The area of the desk top is 14 sticky notes.

Check: count row by row, 5, 10, 12, 14.

Example 3

Part 2: the same placemat in two other units

Avner covers the placemat from Example 1 with his paper cards, each cut exactly as big as two sticky notes side by side. Batsheva covers it with triangles, each made by cutting a sticky note corner to corner. How many cards and how many triangles does each cover take?

Solution

The placemat has an area of 12 sticky notes. Each card is two notes, so the 12 notes can be paired off into cards. Each row of 4 notes makes 2 cards, and there are 3 rows, so 3 \times 2 = 6 cards. Check by pairing: 12 notes in pairs is 6 pairs, since 6 + 6 = 12.

Each note is two triangles, so every note in the cover becomes 2 triangles: 12 \times 2 = 24, or 12 + 12 = 24 triangles.

The placemat is 6 cards or 24 triangles. It is the same area each time. The card is the biggest unit and gives the smallest count; the triangle is the smallest and gives the biggest.

Example 4

Part 2: two picnic blankets with the same trim

Two picnic blankets are sewn from the same square patches. Blanket A is 6 patches long and 2 patches wide. Blanket B is 4 patches long and 4 patches wide. The same length of trim fits around the edge of each. Which blanket covers more ground, and by how much?

Solution

The patches are the same size on both blankets, so they make a fair unit for comparing. Each blanket is an array of patches.

Blanket A has 2 rows of 6 patches: 2 \times 6 = 12 patches.

Blanket B has 4 rows of 4 patches: 4 \times 4 = 16 patches.

Since 16 is more than 12, blanket B covers more ground, by 16 - 12 = 4 patches. The trim around the edges is the same, but the space inside is not: a length around a shape does not decide its area.

Example 5

Part 2: two beach towels in different units

Tomoya says his beach towel has an area of 20 sticky notes. Batsheva measured hers with the cards from Example 3 and says it has an area of 10 cards. Whose towel is bigger?

Solution

The counts 20 and 10 cannot be compared as they stand, because they count different units. Change the cards into notes. Each card is exactly 2 notes, so 10 cards cover the same space as 10 \times 2 = 20 notes.

Batsheva’s towel has an area of 20 sticky notes, the same as Tomoya’s. The towels have equal areas, even though 10 is smaller than 20.

Example 6

Part 3: a strip and a mat

Avner has a paper strip covered by 8 sticky notes in one row, and a mat covered by 2 rows of 4 notes. Without counting, show that the strip and the mat have the same area.

Solution

Cut the strip in half, between the fourth and fifth notes. Each half is 4 notes long, the same as a row of the mat. Slide the right half down and under the left half. Now the two halves sit one above the other, edge to edge, and fit the mat exactly.

Cutting and moving the pieces did not change the strip’s area, and the moved pieces match the mat with no gaps and no overlaps. So the strip and the mat have the same area. Counting agrees: 8 notes and 2 \times 4 = 8 notes.

Example 7

Part 3: tangram pieces measured in small triangles

Batsheva’s tangram set makes the square in the figure. Using a small triangle (piece D or E) as the unit, find the area of each piece, the area of the whole square, and the area of the arrow she makes by sliding piece A to the bottom.

Solution

Each piece can be cut into copies of the small triangle, with nothing left over.

The square piece F, cut from corner to corner, makes 2 small triangles.
The parallelogram G, cut along the short line joining two opposite corners, makes 2 small triangles.
The medium triangle C, cut from its square corner to the middle of its long side, makes 2 small triangles.
Each big triangle, A and B, can be cut into 4 small triangles: its two halves are each the size of the medium triangle, and each of those is 2 small triangles, so 2 + 2 = 4.
D and E are 1 small triangle each.

The pieces fill the square with no gaps and no overlaps, so their areas add to the area of the square:

4 + 4 + 2 + 1 + 1 + 2 + 2 = 16

The square has an area of 16 small triangles. The arrow is the same seven pieces moved, with no overlaps and none missing, so its area is also 16 small triangles.

Boundaries

The unit has to cover without gaps. Coins, buttons or any round shape leave gaps between them however they are packed, so a count of coins on a placemat does not measure its area. Squares, rectangles and triangles cover a flat surface edge to edge; that is why they are used.

Every copy must be the same. A cover made of big and small sticky notes mixed together gives a count that is not in any one unit. The count measures area only when each copy is the same size.

Counts compare only in the same unit. 10 cards and 20 sticky notes can be equal; 10 cards and 15 sticky notes cannot be compared at all until we know how many notes make one card.

Moving pieces keeps the area only if nothing overlaps and nothing is lost. Piling pieces on top of one another covers less space, and leaving a piece out covers less too. Turning a piece over or around is fine.

Area is about flat space. A line has length but covers no space, so it has no area. The room a solid takes up is its volume, measured by filling, as in Volume as filling.

A unit may not fit exactly. Sometimes the copies go right up to the edge and a little space is left that a whole copy cannot fill. Measuring with part units is taken up next, in Measuring area with unit squares.

Where this goes next

Every unit in this lesson has been free to choose, which is exactly why different people get different counts. The next step is to agree on one unit for everyone: a square with sides of one unit of length, such as 1 inch or 1 centimeter. Covering with those squares, counting them on a grid, and counting the part squares at an edge is Measuring area with unit squares. Once the squares sit in rows, the count of a rectangle comes from multiplying its two side lengths, which is Area of rectangles and squares.

Cutting and moving pieces comes back again and again. A slanted parallelogram is cut and slid into a rectangle in Area of parallelograms and trapezoids, an awkward polygon is split into pieces with known areas in Area of any polygon by cutting it into triangles and rectangles, and even a circle is cut into thin wedges and laid out nearly flat in Why the circle area formula works. Each rests on the fact from Part 3: moving the pieces does not change the area.

Filling a solid with copies of a unit cube, instead of covering a flat shape with copies of a flat unit, is Volume as filling.

Exercises

Part 1: covering and counting

A placemat is covered with no gaps and no overlaps by 2 rows of 5 sticky notes. What is its area?
A desk top is covered exactly by 4 rows of 6 sticky notes. What is its area?
A tray is covered exactly by paper triangles, 3 rows with 7 triangles in each row. What is its area, in triangles?
A placemat was covered exactly by 3 rows of 5 sticky notes. Then 2 notes fell off, leaving two gaps, and 13 notes are left on it. What is the area of the placemat in sticky notes?
A shelf top is covered exactly by three rows of sticky notes, with 6, 6 and 3 notes in them. What is its area?
Two placemats, each covered exactly by 8 sticky notes, are laid edge to edge, not overlapping, to make one long mat. What is the area of the long mat?
A desk has 20 sticky notes on it. The notes cover it with no gaps, but 2 of them sit exactly on top of other notes. What is the area of the desk, in sticky notes?

Part 2: units and comparing

A picnic blanket is covered exactly by 9 cards, each as big as two sticky notes side by side. What is its area in sticky notes?
A tray is covered exactly by 10 sticky notes. Each note is then cut corner to corner into 2 triangles. What is the tray’s area in triangles?
A table is covered exactly by 16 sticky notes. What is its area in cards, each the size of two notes?
Two picnic blankets are sewn from the same square patches. Blanket A is 5 patches long and 3 wide. Blanket B is 7 patches long and 1 wide. The same length of trim fits around each one. Which covers more ground, and by how much?
Avner measures his beach towel as 12 cards, each the size of two sticky notes. Batsheva measures hers as 20 sticky notes. Whose towel has the larger area, and by how many sticky notes?
A desk top is covered exactly by 20 sticky notes. A big square sheet of paper is as big as 4 sticky notes in 2 rows of 2. What is the desk’s area in big sheets?
A shape is covered exactly by 8 copies of unit X, or by 32 copies of unit Y. Which unit is bigger? How many copies of Y cover the same space as one copy of X?

Part 3: cutting and rearranging

A strip is covered by 6 sticky notes in one row. It is cut into 3 pieces of 2 notes each, and the pieces are stacked edge to edge into 3 rows of 2. What is the area of the new shape?
In the tangram set of Example 7, what is the area of the two big triangles together, in small triangles? What is the area of the square piece and the parallelogram together?
Batsheva makes a shape from only the two big triangles and the square piece of her tangram set, with no overlaps. What is its area, in small triangles?
A placemat covered exactly by 15 sticky notes is cut into two pieces. One piece is covered by 6 of the notes. What is the area of the other piece?
A shape covered exactly by 8 triangles is cut with scissors into two pieces, and the pieces are put together in a new way with no overlap. What is the area of the new shape, in triangles?
The figure shows three shapes, P, Q and R, each covered by the same triangles. Find the area of each in triangles. Which has the largest area? Do any two have the same area?

Figure 7Three shapes covered by the same triangle, for Exercise 20.

Reasoning

Why can a pile of coins not be used to measure the area of a placemat, even if the coins are all the same size?
Is it true that two blankets with the same length of trim around the edge always cover the same amount of ground? Give an example with square patches.
Tomoya measures a desk top and gets 14 sticky notes. Avner measures the same desk top and gets 7 cards, each the size of two notes. Can both be right? Explain.
Explain why cutting a placemat into pieces and laying the pieces out in a new shape, with no overlaps and none lost, cannot change its area.

Answers

10 sticky notes. Two rows of 5 is 5 + 5 = 10, or 2 \times 5 = 10.
24 sticky notes. Four rows of 6 is 4 \times 6 = 24.
21 triangles. Three rows of 7 is 3 \times 7 = 21.
15 sticky notes. The area of the placemat did not change when notes fell off; it is still the space that 3 \times 5 = 15 notes covered exactly. The 13 notes left do not count the two gaps.
15 sticky notes. The rows add up: 6 + 6 + 3 = 15.
16 sticky notes. The mats do not overlap, so their areas add: 8 + 8 = 16.
18 sticky notes. The 2 notes on top cover space that is already counted, so only 20 - 2 = 18 notes form the cover.
18 sticky notes. Each card is 2 notes, so 9 cards cover the space of 9 \times 2 = 18 notes.
20 triangles. Each note becomes 2 triangles: 10 \times 2 = 20.
8 cards. Pairing 16 notes two at a time makes 8 pairs, since 8 + 8 = 16.
Blanket A, by 8 patches. A has 3 rows of 5, so 3 \times 5 = 15 patches; B has 1 row of 7, so 7 patches; and 15 - 7 = 8.
Avner’s, by 4 sticky notes. His 12 cards are 12 \times 2 = 24 notes, and 24 - 20 = 4.
5 big sheets. Each sheet is 4 notes, and 5 sheets make 4 + 4 + 4 + 4 + 4 = 20 notes.
Unit X is bigger, since fewer copies of it are needed. One X covers the space of 4 copies of Y, because 8 groups of 4 make 8 \times 4 = 32.
6 sticky notes. Cutting and moving the pieces keeps the area; check: 3 \times 2 = 6.
8 small triangles for the two big triangles (4 + 4 = 8), and 4 small triangles for the square piece and the parallelogram (2 + 2 = 4).
10 small triangles: 4 + 4 + 2 = 10.
9 sticky notes. The two pieces together have the placemat’s area, so the other piece is 15 - 6 = 9 notes.
8 triangles. Cutting and putting the pieces together with no overlap and nothing lost keeps the area.
P is 8 triangles, Q is 8 triangles and R is 9 triangles, counted in the figure. R has the largest area. P and Q have the same area, even though one is a square and the other a long slanted shape.
Round coins cannot meet edge to edge. However they are pushed together, gaps are left between them, so some of the placemat is under no coin and the count leaves that space out.
No. A blanket 3 patches by 3 patches and a blanket 5 patches by 1 patch both need trim that is 12 patch edges long, since 3 + 3 + 3 + 3 = 12 and 5 + 1 + 5 + 1 = 12. But the first covers 3 \times 3 = 9 patches and the second only 5.
Yes. They used different units. Each card is 2 notes, so 7 cards cover the space of 7 \times 2 = 14 notes: the same area, counted two ways.
Cover the placemat with sticky notes before cutting. Each piece carries its part of the cover. If the pieces do not overlap, neither do their notes, and if no piece is lost, every note is still there. So the new shape is covered by the same number of notes as the placemat, and has the same area.