Area · Lesson 1 of 12
Understanding Area: Rectangles, Squares, and Perimeter
“Build area from unit squares, distinguish it from perimeter, and solve compound rectangle, path, strip, and scaling problems.”
• Interpret area as the number of square units needed to cover a region without gaps or overlaps. • Use and explain the rectangle formula rather than treating it as a rule to memorise. • Distinguish area from perimeter using examples where one changes while the other does not. • Find areas of paths and compound rectangular regions by subtraction or decomposition. • Reason about rearrangement and scaling, including why doubling a side can quadruple area.
Area begins with a simple question: how much flat surface does a region cover? A drawing may look large because its boundary is long, or compact because its boundary is short, but neither appearance alone tells us its area. To measure surface consistently, we compare it with a standard square unit and ask how many copies of that unit would cover the region exactly.
This lesson develops the rectangle formula from that idea, then uses it in less obvious settings. You will compare perimeter and area, analyse paths around parks, avoid double-counting where paths cross, and see how a bent strip can be rearranged without changing how much surface it covers.
Area and Unit Squares
The measure of the surface enclosed by a two-dimensional region. It is expressed in square units because the measuring unit is a square.
Imagine a rectangle tiled by unit squares, each with side length 1 cm. If 7 squares fit along its length and 4 fit along its width, the rectangle contains 7 rows-by-columns combinations of squares in one direction and 4 in the other, for a total of 7 × 4 = 28 unit squares. Its area is therefore 28 cm².
Problem
One rectangle measures 7 cm by 4 cm. Another measures 8 cm by 3 cm. Which covers more surface?
- 1.First rectangle: 7 × 4 = 28 cm².
- 2.Second rectangle: 8 × 3 = 24 cm².
- 3.The first rectangle covers 4 cm² more, even though the second one has the longer side.
A square is a special rectangle whose length and width are equal. If its side is s, then its area is s × s = s². The exponent 2 is not decoration: it records that two length directions are being multiplied.
Why Perimeter Cannot Measure Area
The total length of the boundary of a closed figure. Perimeter uses length units such as cm or m, while area uses square units such as cm² or m².
Perimeter and area answer different questions. Perimeter asks how far we travel around a boundary. Area asks how much surface lies inside it. Two rectangles can have the same perimeter but different areas, and two figures can have the same area but different perimeters.
Problem
Compare a 1 cm by 9 cm rectangle with a 4 cm by 6 cm rectangle.
- 1.The first perimeter is 2(1 + 9) = 20 cm and its area is 9 cm².
- 2.The second perimeter is 2(4 + 6) = 20 cm and its area is 24 cm².
- 3.The perimeters are equal, but the second region has much greater area.
Problem
Compare a 1 cm by 10 cm rectangle with a 4 cm by 4 cm square.
- 1.The 1 × 10 rectangle has perimeter 22 cm and area 10 cm².
- 2.The 4 × 4 square has perimeter 16 cm and area 16 cm².
- 3.So the figure with the larger perimeter can have the smaller area.
A statement such as “the area is 24 cm” is incomplete because cm is a length unit. Likewise, a perimeter cannot be 24 cm². The unit tells you what kind of quantity you have measured.
Paths Around Rectangles
Suppose a path surrounds a rectangular park. If the outer boundary and inner park dimensions are known, the quickest method is often subtraction. The outer rectangle consists of the park plus the path, so the path is whatever remains after the park area is removed.
Problem
A 20 m by 14 m park has a 2 m wide path outside it on every side. Find the path area.
- 1.The outer dimensions increase by 2 m on both ends of each dimension: 24 m by 18 m.
- 2.Outer area = 24 × 18 = 432 m².
- 3.Park area = 20 × 14 = 280 m².
- 4.Path area = 432 − 280 = 152 m².
A second method is to split the path into non-overlapping rectangles. This is useful when an outer rectangle is not centred around the inner one. The essential rule is that the pieces must cover the path exactly once: no gaps and no overlaps.
Crosspaths and Overlap
When two paths cross inside a plot, adding the area of the horizontal strip and the vertical strip counts the intersection twice. Correct this by subtracting the overlap once. This is the same logic used whenever two regions overlap.
Problem
A 14 m by 12 m plot has a 2 m wide path running through the full 14 m length and another 2 m wide path through the full 12 m width. Find the total path area.
- 1.Lengthwise strip area = 14 × 2 = 28 m².
- 2.Widthwise strip area = 12 × 2 = 24 m².
- 3.The central 2 m × 2 m square was counted twice, so subtract 4 m².
- 4.Total path area = 28 + 24 − 4 = 48 m².
Bent Strips and Rearrangement
A long strip of constant width may bend many times, but bending does not by itself create or destroy area. If the strip can be cut at its turns and rearranged end-to-end without overlap, its total area equals width multiplied by the total effective length. This is a first example of area-preserving rearrangement, an idea used repeatedly later in the chapter.
What Happens When Lengths Scale?
Area responds to changes in length in two directions. If every side of a square is doubled, each horizontal measure doubles and each vertical measure doubles. The new area therefore contains two factors of 2.
Problem
A square has side 5 cm. Its side is doubled. Compare the old and new areas.
- 1.Old area = 5² = 25 cm².
- 2.New side = 10 cm, so new area = 10² = 100 cm².
- 3.100 ÷ 25 = 4, so the area becomes four times as large, not twice as large.
Quiz
What does the area of a region measure?
A rectangle is 9 cm by 4 cm. What is its area?
Two figures have the same perimeter. What must be true about their areas?
When two rectangular paths cross, why may we need to subtract an overlap?
If every side length of a square is doubled, its area is multiplied by
Practice Problems
- A rectangular rangoli region measures 11 cm by 6 cm. Find its area and state the correct unit.
- Find the missing width of a rectangle whose area is 84 m² and length is 12 m.
- Give two different rectangles with perimeter 24 cm but different areas. Calculate both areas.
- A 30 m by 20 m garden has a 1.5 m wide path outside it. Find the path area.
- Two 3 m wide paths cross through the full length and width of a 25 m by 18 m field. Find the total path area.
- A strip 2 cm wide is rearranged from a bent shape into a 45 cm long rectangle. What is its area?
- A square of side 8 cm is enlarged so that every length is three times as large. Find the new area and the area scale factor.
Key Takeaways
• Area counts square units covering a surface; perimeter measures boundary length. • A rectangle has area length × width, and a square has area side². • Equal perimeter does not imply equal area, and larger perimeter does not guarantee larger area. • Compound paths can be handled by subtraction, decomposition, and correcting for overlap. • Rearranging pieces can preserve area even when the outline changes. • When every length is multiplied by a factor k, area is multiplied by k².
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Next · Lesson 2
Area of a Triangle