Lesson 3 of 8 · Grades 3 to 6
Area of rectangles
A rectangle is a tidy grid of unit squares. Counting the rows and the squares in each row gives the rule for area: length times width.
Rows of unit squares
In the last lesson you counted unit squares. A rectangle makes that counting easy, because every row holds the same number of squares. Counting row by row becomes skip counting, and skip counting becomes multiplying.
The rectangle below is 7 cm long and 4 cm wide. It holds 4 rows with 7 unit squares in each row. Instead of counting 28 squares one at a time, work out 4 × 7 = 28. The area is 28 square centimetres, written 28 cm².
The rule for rectangles
The number of squares in a row is the length, and the number of rows is the width, so the rule is short and memorable: area of a rectangle = length × width. Length and width must use the same unit before you multiply. If one is in centimetres and the other in metres, change one of them first.
The answer is always in square units, because you multiplied a length by a length. A rectangle 12 m long and 5 m wide has an area of 12 × 5 = 60 m². A rectangle 8 cm long and 4 cm wide has an area of 8 × 4 = 32 cm².
Squares are rectangles too
A square has four equal sides, so its length and width are the same number. The rule becomes area = side × side. A square with sides of 9 cm has an area of 9 × 9 = 81 cm². A square with sides of 5 m has an area of 25 m².
This is also how you work backwards from an area to a side. If a square has an area of 64 cm², which number times itself gives 64? It is 8, because 8 × 8 = 64. So the side of the square is 8 cm.
Same area, different shape
Different rectangles can cover the same number of squares. An area of 12 square units could be 1 row of 12, 2 rows of 6 or 3 rows of 4. They all look different, but they all hold 12 unit squares. Area counts the surface, not the shape of the outline.
This is why the words matter. Two rectangles with the same perimeter can have different areas. A 1 by 12 rectangle and a 3 by 4 rectangle both use 12 squares, but their perimeters are 26 units and 14 units. Perimeter walks around the edge; area covers the inside.
Finding a missing side
If the area and one side are known, the missing side is found with division. A rectangle with an area of 24 square units and a length of 6 units has a width of 24 ÷ 6 = 4 units. Check it by multiplying: 6 × 4 = 24 gives the area you started with.
Worked example
A rectangle has an area of 45 m² and a width of 9 m. Find the length.
Step 1: write the area rule. Area = length × width, so 45 = length × 9.
Step 2: undo the multiplication with division. 45 ÷ 9 = 5.
Step 3: check the answer. 5 × 9 = 45, which is the area we were given.
Step 4: label the unit. The area was in square metres and the width in metres, so the length is 5 m.
Answer: 5 m.
Watch out for this
Adding the length and the width gives the perimeter, not the area. A rectangle 5 cm by 3 cm has an area of 15 cm², not 8 cm². Also write cm² rather than cm: the little 2 shows that two lengths were multiplied, and it keeps the unit honest.
Remember
- Area of a rectangle = length × width.
- Area of a square = side × side.
- Use the same unit for both lengths, and write the answer in square units.
- To find a missing side, divide the area by the side you know.
Try it
Answer the five questions below. Work them out first, then open each answer to check.
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A rectangle is 5 cm long and 3 cm wide. What is its area?
Show answer
15 cm². 5 × 3 = 15.
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A square has sides of 9 m. What is its area?
Show answer
81 m². 9 × 9 = 81.
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A rectangle has an area of 24 square units and is 6 units long. How wide is it?
Show answer
4 units. 24 ÷ 6 = 4.
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Which area is bigger: a 4 cm by 7 cm rectangle or a 5 cm by 5 cm square?
Show answer
The 4 cm by 7 cm rectangle. Its area is 28 cm², and the square is 25 cm².
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A rectangle is 10 m long and 4 m wide. What is its area?
Show answer
40 m². 10 × 4 = 40.
Puzzle · Missing side
Puzzle time
A rectangular rug has an area of 42 m². Its width is 6 m. How long is the rug?
Show a hint
Area = length × width, so length = area ÷ width.
Show the answer
Answer: 7 m. 42 ÷ 6 = 7, so the rug is 7 m long. Check it: 6 × 7 = 42 m².