Area of rectangles and squares
A game board is ruled into 35 inch squares. Could you know that from its two side lengths alone, without counting a single square?
Where you are 15 in Area and perimeter
Why this exists
Ville is making a game board from a sheet of card. He rules it into squares 1 inch on a side, and it comes out 7 squares across and 5 squares down. To find its area he could count the squares one at a time, as in Measuring area with unit squares. With 35 squares that is slow, and the more squares there are, the easier it is to skip one or count one twice.
But the squares on a rectangle are not scattered about. They sit in rows, every row is the same length, and the rows stack one on another with no gaps. That is an array, from Arrays, and an array is counted by multiplying. Better still, the two numbers of the array are the side lengths themselves: 7 inch squares fit along a side 7 inches long, and 5 rows fit up a side 5 inches long.
So the area of a rectangle can be found with a ruler and one multiplication, with no squares drawn at all. That is how the area of a classroom floor, a garden bed or a sheet of paper is found in practice. This lesson shows why it works, uses it for squares and for real surfaces, and runs it backward once.
Definition
A rectangle is a flat shape with four straight sides and four square corners. Its opposite sides are the same length. The lengths of two sides that meet at a corner are called its length and its width. Which of the two is called the length does not matter.
A square is a rectangle whose four sides are all the same length. So every square is a rectangle, and a rectangle is a square only when its length and its width are equal.
The area of a rectangle means what it meant in Measuring area with unit squares: the number of unit squares that cover it with no gaps and no overlaps. When the sides are measured in inches, the unit square is the square inch; in meters, the square meter. In this lesson both side lengths are whole numbers, measured in the same unit.
Intuition
The lesson runs over four days. Part 1 shows that a rectangle of unit squares is an array whose two numbers are its side lengths. Part 2 finds the same area by tiling and by multiplying, and treats the square. Part 3 measures real surfaces that nobody would tile. Part 4 runs the idea backward once, and asks what happens when the sides grow.
Part 1: a rectangle of unit squares is an array
Here is Ville’s game board, ruled into inch squares.
Look along the bottom side. Each square is exactly 1 inch wide, because a square inch is a square whose sides are 1 inch long. The bottom side is 7 inches long, so exactly 7 squares fit along it, side by side, with nothing left over. That is the bottom row, and every row above it is the same length, so every row has 7 squares.
Now look up the left side. Each row is exactly 1 inch tall, for the same reason. The left side is 5 inches long, so exactly 5 rows fit up it, one on another. So there are 5 rows.
The board is 5 rows of 7, an array, and an array is counted by multiplying the number of rows by the number in each row:
The board is 35 square inches. The side lengths did all the work: one side gave the number of squares in a row, the other gave the number of rows.
That means the squares do not have to be drawn. A rectangle drawn on plain paper, 6 centimeters long and 4 centimeters wide, can be marked at every centimeter along its sides. Joining each mark to the mark across from it cuts the rectangle into square centimeters, and there are 4 rows of 6 of them (Example 2). But the marks only make visible what the side lengths already said. Once that is understood, the lines can be left out and the side lengths multiplied straight away.
Part 2: the same area by tiling and by multiplying, and the square
Tiling and multiplying are not two different answers to compare. They are two ways of counting the same squares, so they always agree. On Ville’s board there are four ways to count:
- one square at a time, 1, 2, 3, \dots, 35;
- by adding the rows, 7 + 7 + 7 + 7 + 7;
- by skip-counting one row at a time, 7, 14, 21, 28, 35;
- by multiplying, 5 \times 7.
All four give 35, because all four count the same 35 squares. Multiplying is simply the quickest.
Turn the board on its side and nothing about its area changes: it covers the same table. Now it has 7 rows of 5, and 7 \times 5 = 35. This is the commutative property seen as area. So the area of a rectangle is length times width or width times length, whichever is easier to work out.
A square is a rectangle with all four sides equal, so everything above holds for it. Soini’s window box has a square base 8 inches on each side.
Each row has 8 inch squares, and there are 8 rows, so the base is 8 \times 8 = 64 square inches. For a square, the number of rows and the number in each row are the same number, the side length. Numbers made this way, 1 \times 1, 2 \times 2, 3 \times 3 and so on, are called square numbers for exactly this reason: they are the areas of squares.
Part 3: real surfaces, measured and not tiled
Nobody covers a classroom floor with paper squares to measure it. A tape measure along two walls that meet at a corner gives the two side lengths, and the reasoning of Part 1 does the rest. The square meters are only imagined: a row of them along one wall, and as many rows as the other wall is meters long.
Two things about units matter here.
Both sides must be in the same unit. The reason length times width works is that each unit of length along a side holds one unit square. If one side is in feet and the other in inches, the “squares” would be 1 foot one way and 1 inch the other, which are not squares at all. One of the lengths has to be changed first, so that both are in feet or both in inches.
The answer is in square units. Sides in meters give square meters; sides in feet give square feet. A floor is not “54 meters” any more than it is 54 pencils. An area and a length are never added together.
Some surfaces are long. A school hallway 3 meters wide and 20 meters long holds 3 rows of 20 square meters. Each row of 20 is 2 tens, so the hallway is 3 rows of 2 tens, which is 3 \times 2 = 6 tens, or 60 square meters (see Multiplying by tens and hundreds).
Because an area is a count of square units, areas in the same unit can be added and subtracted like any counts: two boards together, or the part of a floor a rug leaves bare (Example 6).
Part 4: working backward, and changing the sides
Sometimes the area is known and one side is missing. Picture the unit squares in rows along the known side: each row holds as many squares as that side is units long, and is 1 unit deep. The question “how long is the other side?” becomes “how many rows make the area?”, which is a division (Example 7). Division here asks how many equal groups, one of the two questions in What division means. The answer is a length, in feet, not square feet. This backward step gets a full lesson of its own in Finding a missing side from the area.
The picture of rows also says what happens when a side grows. Make a rectangle twice as long and each row holds twice as many squares, so the area doubles. Make it twice as long and twice as wide, and there are twice as many rows, each twice as long: the area is not doubled but made four times as large (Example 8).
Results
The area of a rectangle. A rectangle whose length is l units and whose width is w units, both whole numbers in the same unit, has area
The area of a square. A square whose side is s units has area s \times s square units.
Either order. l \times w = w \times l, so either side may be written first.
A missing side. If a rectangle’s area and one side are known, the other side is the area divided by the known side, and it is a length, in the same unit as the known side.
Why it is true
Take a rectangle whose length is l units and whose width is w units, both whole numbers.
Mark the bottom side at every unit, which cuts it into l pieces each 1 unit long, and mark the left side the same way into w pieces. Through each mark on the bottom, draw a line straight up to the top side; through each mark on the left, draw a line straight across to the right side. The corners of the rectangle are square corners, so the lines going up and the lines going across meet at square corners too, and they cut the whole rectangle into small pieces, each 1 unit wide and 1 unit tall. Each piece is a unit square. The lines only cut the rectangle, so these squares cover it with no gaps and no overlaps, and the area of the rectangle is the number of them.
The strip between two neighboring lines going across is one row. It is 1 unit tall and l units long, and the lines going up cut it into l unit squares. There are w such strips, one for each piece of the left side, and every one of them holds the same l squares. So the squares form an array of w rows of l, which holds w \times l squares. By the commutative property this is l \times w.
A square is the case where l and w are the same number s, so its area is s \times s.
The argument used the square corners twice: to make the pieces squares, and to make every row the same length. That is why it proves nothing about a shape without square corners. \blacksquare
Worked examples
Examples 1 and 2 go with Part 1, Examples 3 and 4 with Part 2, Examples 5 and 6 with Part 3, and Examples 7 and 8 with Part 4.
Ville’s game board
Ville’s game board is a rectangle 7 inches long and 5 inches wide, ruled into squares 1 inch on a side. Find its area by adding the rows, and again by multiplying.
Each small square is 1 inch on a side, so each is 1 square inch, and together they cover the board with no gaps.
Along the 7-inch side, 7 squares fit, because each one is 1 inch wide. So each row has 7 squares. Each row is 1 inch tall, so the 5-inch side holds 5 rows. Adding one 7 for each row:
The board is 5 equal rows of 7, so the same count is a multiplication:
The area is 35 square inches. Check by skip-counting one row at a time: 7, 14, 21, 28, 35, five numbers for five rows.
Joining the ruler marks
Kaisu draws a rectangle on plain paper, 6 centimeters long and 4 centimeters wide. She makes a mark at every centimeter along all four sides and joins each mark to the one across from it with a dashed line. What shapes do the dashed lines cut the rectangle into, and what is its area?
The marks cut the long sides into 6 pieces each 1 centimeter long, and the short sides into 4 pieces each 1 centimeter long. The corners of a rectangle are square, so the dashed lines going up and the dashed lines going across meet at square corners. Each piece they make is 1 centimeter wide and 1 centimeter tall: a square centimeter.
There is one row for each centimeter of the short side, 4 rows, and each row has one square for each centimeter of the long side, 6 squares:
The rectangle is 24 square centimeters. Kaisu did not need the dashed lines to know this: the side lengths alone said 4 rows of 6. Check by columns instead: 6 columns of 4, and 6 \times 4 = 24.
Soini’s window box
Soini’s window box has a square base, 8 inches on each side. What is the area of the base?
A square is a rectangle with all its sides equal, so its area is length times width, and here both are 8 inches. Inch squares sit in rows of 8 along one side, and the other side holds 8 rows:
If 8 \times 8 is not yet a known fact, split the 8 rows into two groups of 4 rows. Each group is 4 \times 8 = 32 squares, and 32 + 32 = 64.
The base is 64 square inches.
Which board covers more?
Kaisu has two game boards. One is a square 5 inches on each side. The other is a rectangle 8 inches long and 3 inches wide. Which covers more of the table, and by how much?
SolutionThe square holds 5 rows of 5 inch squares:
The rectangle holds 3 rows of 8:
The square board covers 25 square inches and the long board 24, so the square covers more, by 25 - 24 = 1 square inch. The long board looks bigger because one of its sides is longer, but area depends on both sides at once.
Check by adding rows, the tiling count: 5 + 5 + 5 + 5 + 5 = 25 and 8 + 8 + 8 = 24.
Ville’s classroom floor
Ville’s classroom floor is a rectangle 9 meters long and 6 meters wide. What is its area?
SolutionImagine the floor covered in squares 1 meter on a side. Along the 9-meter wall, 9 of them fit in a row. Each row is 1 meter deep, so the 6-meter wall holds 6 rows. Skip-count by nines, one number per row: 9, 18, 27, 36, 45, 54. So
Both sides are in meters, so the unit square is the square meter. The floor is 54 square meters.
The floor the rug leaves bare
A rug 5 meters long and 3 meters wide lies on Ville’s classroom floor from Example 5. How much of the floor does the rug leave uncovered?
The rug is a rectangle, so its area is
square meters. The floor is 54 square meters, and the rug covers 15 of those square meters. The part left bare is what remains:
Taking one area from another is allowed here because both are counts of the same unit, square meters. The uncovered floor is 39 square meters. Check: 39 + 15 = 54.
The game board on the playground
Kaisu is painting a rectangular game board on the playground. It must cover 30 square feet, and its side along the fence is 6 feet long. How long is the other side?
Picture the board’s square feet in rows along the fence. The fence side is 6 feet long, so each row holds 6 square feet, and each row is 1 foot deep. The other side is as many feet long as there are rows. So the question is how many rows of 6 make 30:
Count up in sixes, one number per row: 6, 12, 18, 24, 30. That is 5 rows, so 30 \div 6 = 5.
Five rows, each 1 foot deep, make a side 5 feet long. The answer is a length, so it is 5 feet, not 5 square feet. Check: 6 \times 5 = 30.
Twice as long and twice as wide
Soini’s small window box has a base 4 inches long and 3 inches wide. A bigger box has a base with both sides twice as long. How many times the area of the small base is the big one?
SolutionThe small base is
square inches. The big base is 8 inches long and 6 inches wide, so it is
square inches. Count how many twelves make 48: 12, 24, 36, 48. That is 4, so the big base is 4 times the small one, not 2 times.
The rows show why. Doubling the length makes every row twice as long, which doubles the area. Doubling the width as well makes twice as many rows, which doubles it again. Twice and then twice again is 4 times. In a picture, the big base holds 4 copies of the small one, 2 across and 2 up.
Boundaries
Square corners are needed. Length times width is the area only of a rectangle. A slanted four-sided shape with sides 5 and 3 units long does not cover 15 square units: lines through its side marks do not cut it into squares, and pushing a rectangle over into a slant makes it flatter, with less inside. Shapes like that are measured in Area of parallelograms and trapezoids.
Every row must be the same length. A shape whose rows have different lengths, such as an L shape, is not a rectangle, and no single product of two side lengths gives its area. It can be cut into rectangles whose areas are added, which is Splitting a figure into rectangles.
Both sides in the same unit. A side in feet and a side in inches must be put into one unit before multiplying.
The answer is in square units, and a missing side is in plain units. Multiplying two lengths gives an area; dividing an area by a length gives a length.
Whole-number sides. The argument here cuts each side into whole units. A side such as \displaystyle 2\frac{1}{2} inches leaves part squares along the edge; length times width still gives the area, but showing why needs fractions, in A fraction times a fraction.
Where this goes next
The rectangle is the picture behind a great deal of multiplication. In the area model, the side lengths are the factors and the area is the product, and cutting one rectangle into two shows how a hard product splits into easier ones.
Shapes made of rectangles, like an L-shaped room, are found by splitting them into rectangles and adding, in Splitting a figure into rectangles, and later shaded regions are found by subtracting, in Area of composite figures and shaded regions. Working backward from an area to a side, met once in Part 4, is the whole subject of Finding a missing side from the area.
The distance around a rectangle, measured in plain units, is its Perimeter, a different measure of the same shape. And the same row-by-row reasoning, one level up, stacks layers of cubes in a box: Volume with unit cubes.
Exercises
Exercises 1 to 6 go with Part 1, 7 to 12 with Part 2, 13 to 17 with Part 3, and 18 to 21 with Part 4. The reasoning group draws on all four.
Part 1: a rectangle of unit squares is an array
The figure shows rectangles A and B, for Exercises 1 and 3.
- Rectangle A is ruled into centimeter squares. How many squares are in each row, how many rows are there, and what is its area?
- A score strip for a game is a rectangle 9 inches long and 2 inches wide, ruled into inch squares. What is its area?
- Rectangle B has a mark at every centimeter along its sides and no lines inside. Find its side lengths from the marks, and its area.
- Kaisu covers a game board with square stickers 1 inch on a side, with no gaps. She puts 9 stickers in each row and makes 3 rows. What is the area of the board, and what are its side lengths?
- A rectangle is 7 centimeters long and 1 centimeter wide. What is its area?
- A rectangle is 6 meters long and 2 meters wide. Write its area as an addition of rows and as a multiplication, and find it.
Part 2: tiling and multiplying, and the square
- A square window box base is 6 inches on each side. What is its area?
- A square tile is 9 centimeters on each side. What is its area?
- Ville covers a rectangle with centimeter squares and counts 32 of them. With a ruler he measures the rectangle as 8 centimeters long and 4 centimeters wide. Does multiplying agree with his count? Explain why it must.
- Which covers more, a square 7 inches on each side or a rectangle 8 inches long and 6 inches wide? By how much?
- One rectangle is 2 feet long and 8 feet wide; another is 8 feet long and 2 feet wide. Compare their areas.
- A square game board is 10 inches on each side. What is its area?
Part 3: real surfaces
- Soini’s window box base is 9 inches long and 7 inches wide. How many square inches of lining cover it exactly?
- A school hallway is 4 meters wide and 20 meters long. What is the area of its floor?
- Kaisu paints a game board 8 feet long and 5 feet wide on the playground. One can of paint covers 8 square feet. How many cans does she need?
- One classroom floor is 8 meters by 7 meters. Ville’s is 9 meters by 6 meters. Which floor is larger, and by how much?
- Soini has two game boards, one 5 inches by 4 inches and one 6 inches by 3 inches. How much of the table do they cover together, if they do not overlap?
Part 4: working backward, and changing the sides
- A rectangle has an area of 18 square centimeters, and one side is 3 centimeters. How long is the other side?
- A square window box base has an area of 49 square inches. How long is each side?
- A game board is 5 inches long and 2 inches wide. A second board has both sides twice as long. Find both areas. How many times the first is the second?
- Kaisu has 20 square tiles, each 1 foot on a side. List every rectangle with whole-foot sides she can make using all of them.
Reasoning
- A rectangle is 9 centimeters long and 5 centimeters wide. Without drawing any squares, explain why 9 tells how many square centimeters are in each row and 5 tells how many rows there are.
- Is every square a rectangle? Is every rectangle a square? Explain.
- A shape is made of rows of 5, 5 and 3 inch squares, lined up at the left. It is 5 inches across at its widest and 3 inches tall. Explain why 3 \times 5 does not give its area, and find the area.
Answers
- 20 square centimeters. Each row has 5 squares and there are 4 rows: 4 \times 5 = 20.
- 18 square inches. Two rows of 9: 2 \times 9 = 18.
- The marks show sides of 7 centimeters and 4 centimeters, so 4 rows of 7: 4 \times 7 = 28 square centimeters.
- 27 square inches: 3 \times 9 = 27. The board is 9 inches by 3 inches, one inch of side for each sticker along it.
- 7 square centimeters. One row of 7: 1 \times 7 = 7.
- 12 square meters. Two rows of 6: 6 + 6 = 12, and 2 \times 6 = 12.
- 36 square inches: 6 \times 6 = 36.
- 81 square centimeters: 9 \times 9 = 81.
- Yes: 4 \times 8 = 32. It must agree, because the 4 rows of 8 that multiplying counts are the very squares he counted one by one.
- The square, by 1 square inch. The square is 7 \times 7 = 49 and the rectangle is 6 \times 8 = 48 square inches, and 49 - 48 = 1.
- They are equal, 16 square feet each: 2 \times 8 = 16 and 8 \times 2 = 16. The second is the first turned on its side.
- 100 square inches: 10 \times 10 = 100.
- 63 square inches: 7 \times 9 = 63.
- 80 square meters. Four rows of 20, each 2 tens, is 4 \times 2 = 8 tens, which is 80.
- 5 cans. The board is 5 \times 8 = 40 square feet, and 40 \div 8 = 5.
- The 8 by 7 floor, by 2 square meters. It is 8 \times 7 = 56 square meters against 9 \times 6 = 54, and 56 - 54 = 2.
- 38 square inches. The boards are 4 \times 5 = 20 and 3 \times 6 = 18, and 20 + 18 = 38.
- 6 centimeters. Rows of 3 along the known side: \square \times 3 = 18, and 18 \div 3 = 6.
- 7 inches, since 7 \times 7 = 49. (6 \times 6 = 36 is too small and 8 \times 8 = 64 too big.)
- 10 square inches and 40 square inches. The first is 2 \times 5 = 10; the second is 10 inches by 4 inches, so 4 \times 10 = 40. That is 4 times the first, since 4 \times 10 = 40.
- 1 foot by 20 feet, 2 feet by 10 feet, and 4 feet by 5 feet (and each of these turned on its side). Rows of 3 fail (6 \times 3 = 18, 7 \times 3 = 21), and so do rows of 6, 7, 8 and 9.
- A square centimeter is 1 centimeter wide, so along the 9-centimeter side exactly 9 of them fit in a row. Each row is 1 centimeter tall, so the 5-centimeter side holds exactly 5 rows. The area is 5 \times 9 = 45 square centimeters.
- Every square is a rectangle: it has four straight sides and four square corners. Not every rectangle is a square: a square also needs all four sides equal, and a rectangle 8 by 3 has unequal sides.
- 13 square inches: 5 + 5 + 3 = 13. The product 3 \times 5 = 15 counts 3 full rows of 5, but the shape’s last row is short by 2 squares, so it is not a rectangle and the product counts 2 squares that are not there: 15 - 2 = 13.