Count the squares
Area counts unit squares. A rectangle 2 units wide and 3 units tall holds 3 rows of 2 squares, so its area is 2 × 3 = 6 squares.
A rectangle 2 squares wide and 3 squares tall holds 6 unit squares.
Double every side
Now double every side. The width becomes 2 × 2 = 4 and the height becomes 3 × 2 = 6, and the new rectangle holds 4 × 6 = 24 squares. That is 4 times as many squares as before, not 2 times.
The area grows 4 times because it grows in two directions. Doubling the width doubles the number of squares in each row, and doubling the height doubles the number of rows. Twice as many rows, each twice as long, is 2 × 2 = 4 times as many squares.
The 4 by 6 rectangle holds 6 rows of 4, which is 24 unit squares. The colored corner is the 2 by 3 rectangle: 2 of it fit across and 2 fit up, so 2 × 2 = 4 of it fit inside.
Each direction picks up its own factor
The same happens for any scale factor. Multiply every side by k, and the width w becomes k × w while the height h becomes k × h. The area becomes (k × w) × (k × h) = k × k × w × h, so the area is multiplied by k × k.
Doubling the sides multiplies the area by 2 × 2 = 4. Tripling them multiplies it by 3 × 3 = 9: the 2 by 3 rectangle becomes 6 by 9, which holds 54 squares, and 54 = 9 × 6.
Tripled, the rectangle is 6 by 9 and holds 54 unit squares. The colored 2 by 3 corner fits 3 times across and 3 times up, so 3 × 3 = 9 of it fit inside.
length ×1, area ×1, volume ×1
Make the area scale factor 9
Drag the corner to make the scale factor k equal to 2, 3 or 4. The length of the line is multiplied by k. The square is cut into k rows of k unit squares, so its area is multiplied by k × k. Each face of the cube is ruled k by k, and the cube holds k layers of them, so its volume is multiplied by k × k × k.
Areas on plans and maps
The same rule works on a plan or a map. On a floor plan drawn to the scale 1 : 50, every length on the floor is 50 times its length on the plan. So every area on the floor is 50 × 50 = 2,500 times its area on the plan. A 1 cm square on the plan stands for a piece of floor 50 cm by 50 cm, which is 2,500 cm².
The same is true of a cake tin. A square tin with sides twice as long holds 2 × 2 = 4 times as much batter, filled to the same depth.
The usual mistakes
Multiplying the area by the scale factor only once. When the sides of the 2 by 3 rectangle double, the area is not 2 × 6 = 12 squares. The width and the height both double, so the area is 2 × 2 × 6 = 24 squares.
Adding the two factors. The width factor multiplies the height factor. Tripling the sides multiplies the area by 3 × 3 = 9, not by 3 + 3 = 6.
Giving every side its own factor. A rectangle has four sides but only two directions, across and up. Doubling multiplies the area by 2 × 2 = 4, not by 2 × 2 × 2 × 2 = 16.
Worked example: An L-Shaped Living Room on a Floor Plan, and the Boxes of Flooring It Needs
Question A floor plan of an apartment is drawn to the scale 1 : 50 on paper ruled in 1 cm squares. The L-shaped living room covers 108 of the squares. (a) What is the floor area of the real living room, in square meters? (b) Flooring is sold in boxes that each cover 2.2 m2, at $64 a box. How much does it cost to buy enough boxes to cover the living room floor? Ignore any waste from cutting.
1.One 1 cm square on the plan stands for a square of floor 50 cm by 50 cm, which is 0.5 m by 0.5 m.
One 1 cm square of the plan stands for 50 cm by 50 cm of floor: 0.5 m by 0.5 m. 2.That piece of floor has an area of 0.5 × 0.5 = 0.25 m2. Both of its sides were scaled by 50, so its area was scaled by 50 × 50 = 2500.
Its area is 0.5 × 0.5 = 0.25 m2. Both sides were scaled by 50, so the area is scaled by 2500. 3.(a) The room covers 108 squares, so its area is 108 × 0.25 = 27 m2.
(a) 108 squares: 108 × 0.25 = 27 m2. 4.(b) 12 boxes cover 12 × 2.2 = 26.4 m2, which is less than 27 m2, so 13 boxes are needed.
(b) 12 boxes cover 26.4 m2, short of 27 m2, so 13 boxes are needed. 5.13 boxes cost 13 × $64 = $832. Check: 13 × 2.2 = 28.6 m2, enough for the 27 m2 room.
13 × $64 = $832.
Answer: (a) 27 m2; (b) $832
Common mistakes
- Multiplying the plan area by 50 only: 108 × 50 = 5400 cm2, about half a square meter, far too small for a living room. The scale multiplies each length by 50, and an area is two lengths multiplied, so it is multiplied by 50 × 50 = 2500.
- Rounding 12.27… boxes down to 12. Twelve boxes cover 26.4 m2 and leave 0.6 m2 of floor bare, so a thirteenth box must be bought.