11.6 Area by Tiling, Unit Squares, & the Distributive Property of Area

Area by Tiling, Unit Squares, & the Distributive Property of Area

Section 11.3 gave you the formulas. This section gives you the reasoning behind them, because Levels E and M assess why $A = lw$ works and use that reasoning in items where no formula applies directly.

Area Is a Count of Unit Squares

Area is the number of unit squares — squares 1 unit on a side — needed to cover a figure with no gaps and no overlaps.

That definition explains the unit. A square 1 foot on each side has an area of 1 square foot, written 1 sq ft or 1 ft². Cover a shape with 24 of them and its area is 24 square feet.

A 5 × 3 rectangle tiled with unit squares:

  ┌───┬───┬───┬───┬───┐
  │ 1 │ 2 │ 3 │ 4 │ 5 │
  ├───┼───┼───┼───┼───┤
  │ 6 │ 7 │ 8 │ 9 │10 │
  ├───┼───┼───┼───┼───┤
  │11 │12 │13 │14 │15 │
  └───┴───┴───┴───┴───┘

  3 rows × 5 squares per row = 15 unit squares

The formula is a counting shortcut. Rather than counting 15 squares one at a time, notice there are 3 identical rows of 5, and multiply: $A = l \times w = 5 \times 3 = 15$. That is all $A = lw$ has ever meant.

Why area units are "square." Multiplying feet by feet produces $\text{ft} \times \text{ft} = \text{ft}^2$. If your answer's unit is not squared, you did not compute an area.

Counting on a Grid

TABE shows figures drawn on grid paper and asks for the area.

Worked estimate. A curved region covers 14 whole squares and clips 8 partial squares. Estimated area $\approx 14 + (8 \times \tfrac{1}{2}) = \mathbf{18}$ square units.

The Distributive Property of Area

A rectangle can be cut into two smaller rectangles by a single straight line, and the areas add:

   ┌─────────────┬───────┐
   │             │       │
 4 │   4 × 10    │ 4 × 3 │
   │    = 40     │  = 12 │
   └─────────────┴───────┘
         10          3
      Total: 4 × 13 = 52 = 40 + 12

$4 \times 13 = 4 \times (10 + 3) = (4 \times 10) + (4 \times 3) = 40 + 12 = 52$

This is the distributive property, and the picture is the proof. It is also the fastest mental method for awkward multiplications on Part 1, where no calculator is available: $7 \times 26 = 7(20 + 6) = 140 + 42 = 182$.

Decomposition in reverse

Any L-shaped figure splits into two rectangles. Cut it, compute both, add:

   ┌──────┐
   │      │  6
   │      │
   ├──────┴───────┐
   │              │  5
   └──────────────┘
      4      +   9

Top rectangle: $4 \times 6 = 24$. Bottom rectangle: $13 \times 5 = 65$. Total: $\mathbf{89}$ square units. Section 11.3 covers the subtraction method for the same figures — either works, and adding is usually safer because it never requires inferring a hidden dimension.

Composing to Find Triangle Area

Why is a triangle's area $\frac{1}{2}bh$? Because two identical triangles compose into a parallelogram, and a parallelogram shears into a rectangle of the same base and height.

   Triangle           Two copies form           Same area as
   b × h              a parallelogram           a b × h rectangle
       /|                  /|‾‾‾/                ┌────────┐
      / |                 / |  /                 │        │
     /  | h              /  | /  h               │        │ h
    /___|               /___|/                   └────────┘
      b                    b                          b

   Area = ½ × b × h        Area = b × h            Area = b × h

One triangle is therefore half of $b \times h$. The same composing argument gives the trapezoid formula, which is why $A = \frac{1}{2}(b_1 + b_2)h$ — the average of the two bases, times the height.

Same Area, Different Perimeter

This comparison appears on TABE regularly, and the answer surprises people.

All of these rectangles have area 36:

Same area, wildly different perimeters. The square has the smallest perimeter of all rectangles with a given area — the practical reason a square pen uses the least fencing.

The relationship runs the other way too. All of these have perimeter 24:

Same perimeter, different areas — and again the square maximizes.

The takeaway TABE tests: knowing one of area or perimeter does not determine the other. Any item that asks you to find perimeter from area alone, without dimensions, is unanswerable — and TABE writes exactly that kind of reasoning question.