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Area by counting squares

Measurement · level 3, lesson 4 of 6

A question to start

You have 12 metres of fence to make a pen for a rabbit, and you want the pen to be a rectangle. You could make it long and thin, 5 m by 1 m. Or you could make it a square, 3 m by 3 m. Both use all 12 m of fence: \(5 + 1 + 5 + 1 = 12\) and \(3 + 3 + 3 + 3 = 12\).

Does the rabbit get the same amount of grass to eat in both pens? Most people guess "yes, the fence is the same". But the square pen has almost twice as much grass. The fence measures the edge; the grass is about the space inside. That space is called area, and in this lesson you learn to measure it.

The idea

What area is

Area is the amount of flat surface something covers. The top of a table has an area. So does a page, a floor, a football field and a country. Area answers the question "how much surface?", just as length answers "how long?".

To measure length, we lay rulers or steps end to end and count. To measure area, we cover the surface with equal tiles and count the tiles.

Square units

Any equal tiles would do, but squares are the best choice, because squares fit together with no gaps, row after row, like the tiles on a bathroom floor. So area is measured in square units.

  • A square centimetre (cm²) is a square 1 cm long and 1 cm wide, about the size of your fingernail. Good for measuring a stamp, a photo or a page.
  • A square metre (m²) is a square 1 m long and 1 m wide. Four sheets of newspaper taped together make about one square metre. Good for measuring a floor, a wall or a garden.
  • A square kilometre (km²) is a square 1 km long and 1 km wide, used for parks, cities and countries.

The little raised 2 in cm² is read "squared". It is a reminder that a square has two directions, a length and a width. "Square centimetre" and "cm²" mean exactly the same thing.

Do not mix up a square centimetre with a centimetre. A centimetre is a length: a short line. A square centimetre is an area: a little patch of surface.

Rules for counting

When you count squares to find an area, the same honest rules apply as when you measured length:

  1. No gaps. Every bit of the shape must be covered.
  2. No overlaps. No bit may be covered twice.
  3. All the same size. Every tile must be the same square unit.

Half squares

Some shapes cut squares in half along a diagonal (corner to corner). A diagonal cuts a square into two triangles that are exactly the same, so each one is half a square. Two halves make one whole.

6 whole squares 4 half squares 6 + 2 = 8 squares

This triangle covers 6 whole squares and 4 half squares. Four halves make 2 wholes, so the area is \(6 + 2 = 8\) square units.

Here is a way to check: the triangle is exactly half of the 4 by 4 square around it (the slanted side cuts that square corner to corner). The 4 by 4 square has 16 squares, and half of 16 is 8. Both ways agree. Remember this idea: in level 4 it becomes the rule for the area of every triangle.

Curvy shapes: estimating

A circle, a leaf or the outline of your hand does not fit the grid exactly. Then we estimate. Here is a circle drawn on a grid of 1 cm squares.

16 whole 16 more than half 4 corners: tiny bits
  • Too small: count only the whole squares inside: 16 cm². The real area must be more than this.
  • Too big: count every square the circle touches at all: 36 cm². The real area must be less than this.
  • A good estimate: count the whole squares, then count each square that is more than half inside as one, and leave out the ones that are less than half inside. That gives \(16 + 16 = 32\) cm².

So the area is somewhere between 16 and 36 cm², and about 32 cm². The true area of this circle is about 28 cm² (you will learn how to work that out exactly in Geometry). Our estimate is close, but not perfect. With a finer grid, say squares of half a centimetre, the bits we guess about get smaller and the estimate gets better. Mathematicians use this very idea, with smaller and smaller squares, to find exact areas of curved shapes.

Same area, different perimeter

The perimeter (from level 2) is the distance all the way around the edge. On a grid you can count it by counting the edges of squares along the outside.

Here are two shapes, each made of the same 6 squares.

area 6, perimeter 10area 6, perimeter 14

Both have area 6 square units, but the 3 by 2 rectangle has perimeter \(3 + 2 + 3 + 2 = 10\) units, and the 6 by 1 strip has perimeter \(6 + 1 + 6 + 1 = 14\) units.

Why can the perimeter change when the area does not? Every square has 4 edges. When two squares sit side by side, the edge they share is inside the shape, so it is not part of the perimeter: two outside edges are lost. The rectangle has its squares packed tightly, with 7 shared edges, so it loses a lot: \(6 \times 4 - 2 \times 7 = 10\). The strip has only 5 shared edges: \(24 - 10 = 14\). Compact shapes have short edges; stretched shapes have long edges.

The same idea works the other way: the same perimeter can surround different areas, which is the rabbit puzzle.

Worked examples

1. Counting. A shape on a centimetre grid covers 9 whole squares and 6 half squares. Its area?

6 halves make 3 wholes, so the area is \(9 + 3 = 12\) cm².

2. The rabbit pens. Draw each pen on squared paper with one square for each square metre.

  • The 5 m by 1 m pen has 1 row of 5 squares: 5 m².
  • The 3 m by 3 m pen has 3 rows of 3 squares: 9 m².

Both have perimeter 12 m. The square pen has 4 m² more grass.

3. All the rectangles with area 12. Find every rectangle made of 12 whole squares, and its perimeter.

The rows must share the 12 squares equally: 1 row of 12, 2 rows of 6, or 3 rows of 4 (4 rows of 3 is the same rectangle turned round).

Rectangle Area Perimeter
12 by 1 12 \(12 + 1 + 12 + 1 = 26\)
6 by 2 12 \(6 + 2 + 6 + 2 = 16\)
4 by 3 12 \(4 + 3 + 4 + 3 = 14\)

The closer the rectangle is to a square, the shorter its perimeter.

4. Choosing the unit. Which unit would you use for the area of: a postage stamp; a classroom floor; a lake?

Stamp: cm² (about 6 cm²). Classroom floor: m² (perhaps 60 m²). Lake: km².

5. A hand. Trace your hand on centimetre squared paper. Count the whole squares inside, then add the squares that are more than half inside. Many 10-year-olds get something near 100 cm², give or take a few dozen. Try it with a grown-up's hand and compare: you will both be estimating, so you will also learn how close two estimates can be.

Try it

  1. A shape covers 14 whole squares and 4 half squares. What is its area?
  2. On squared paper, draw two different shapes with area 5 squares. Find the perimeter of each.
  3. A rectangle on a grid is 4 squares long and 2 squares wide. What are its area and perimeter?
  4. Find all the rectangles made of 16 whole squares. Which has the shortest perimeter?
  5. A right triangle on a grid has its two short sides along grid lines, 6 squares and 6 squares long. Its slanted side cuts squares corner to corner. What is its area? (Hint: what is it half of?)
  6. Can two shapes have the same perimeter but different areas? Give an example on a grid.
  7. A kitchen floor is covered by tiles that are each 1 m by 1 m: 4 rows of 5 tiles. What is the area of the floor? If every tile were cut into 4 smaller squares of 50 cm by 50 cm, how many small tiles would there be?
Answers
  1. 4 halves make 2 wholes, so \(14 + 2 = 16\) square units.
  2. Many answers. A row of 5 squares has perimeter \(5 + 1 + 5 + 1 = 12\). A plus sign (one square with a square on each side) also has area 5, and its perimeter is also 12 (count the outside edges: each of the 4 arms shows 3 edges). An L-shape (3 in a row with 2 more on top of one end) has area 5 and perimeter 12 as well. A 2 by 2 square with one extra square stuck on has perimeter 10. Compact shapes have shorter perimeters.
  3. Area \(4 \times 2 = 8\) squares (2 rows of 4). Perimeter \(4 + 2 + 4 + 2 = 12\) units.
  4. 16 by 1 (perimeter 34), 8 by 2 (perimeter 20) and 4 by 4 (perimeter 16). The 4 by 4 square has the shortest perimeter.
  5. It is half of a 6 by 6 square, which has 36 squares. So its area is \(36 \div 2 = 18\) squares. (Counting: 15 whole squares and 6 half squares, \(15 + 3 = 18\). ✓)
  6. Yes. A 5 by 1 rectangle and a 3 by 3 square both have perimeter 12, but areas 5 and 9.
  7. 4 rows of 5 tiles: \(4 \times 5 = 20\) m². Each big tile becomes 4 small tiles, so \(20 \times 4 = 80\) small tiles. (Same floor, same area: only the unit changed.)

Common mistake

Mixing up area and perimeter. Perimeter is a length (the fence), counted in cm or m. Area is a surface (the grass), counted in cm² or m². If a question asks "how much carpet?" it wants area; "how much skirting board around the edge?" wants perimeter.

Counting half squares as whole ones. When a slanted line cuts squares corner to corner, each piece is only half. Pair the halves up before you count.

Where it leads