Measuring Area by Counting Unit Squares - Third Grade Mathematics

While perimeter measures the boundary distance around a shape, area measures the flat surface space inside that boundary. When tiling a bathroom floor, laying down a soft rug in a bedroom, or painting a canvas, you need to know how much surface must be covered. In this chapter, you will discover the concept of area, explore unit squares as the universal measuring blocks of area, and count unit squares to find the area of figures without gaps or overlaps.

What Is Area?

Area is the amount of two-dimensional surface space covered by a flat figure. Think of the difference between perimeter and area like a picture frame versus the photograph inside:
- Perimeter is the wooden frame running around the edge.

- Area is the flat glossy photograph filling the space inside!


+-----------------------+---------------------+
| Perimeter             | Area                |
+-----------------------+---------------------+
| Outside boundary      | Inside surface      |
| Distance walked       | Space covered       |
| Measured in units     | Measured in         |
| (cm, inches, feet)    | SQUARE units        |
|                       | (sq cm, sq inches)  |
+-----------------------+---------------------+

What Is a Unit Square?

The standard building block of area is a unit square. A unit square is a square whose sides all have a length of 1 unit.

       1 unit
      +-----+
1 unit|     | 1 unit
      +-----+
       1 unit
Area = 1 Square Unit (sq unit)

Common standard unit squares include:
- 1 square inch (a square measuring 1 inch on each side)

- 1 square centimeter (a square measuring 1 cm on each side)

- 1 square foot (a square floor tile measuring 1 foot on each side)

- 1 square meter (a large square measuring 1 meter on each side)


The Fundamental Rules of Measuring Area by Tiling

To measure the area of any shape using unit squares, the tiles must follow two non-negotiable geometric laws:
1. No Gaps: There can be no empty spaces between the unit squares.

2. No Overlaps: No unit square can be placed on top of another unit square.


Correct Tiling (Valid Area Measurement):
+---+---+---+---+
| 1 | 2 | 3 | 4 |
+---+---+---+---+
| 5 | 6 | 7 | 8 |
+---+---+---+---+
Area = 8 square units! Perfect grid with no holes and no stacked tiles.

Counting Unit Squares in Regular and Irregular Shapes

You can find the area of any flat shape on a grid simply by counting the total number of unit squares it covers!

Example 1: Rectangle on a Grid

+---+---+---+---+---+
| 1 | 2 | 3 | 4 | 5 |
+---+---+---+---+---+
| 6 | 7 | 8 | 9 | 10|
+---+---+---+---+---+
| 11| 12| 13| 14| 15|
+---+---+---+---+---+
Count = 15 unit squares.
Area = 15 square units.

Example 2: L-Shaped Figure on a Grid

+---+---+
| 1 | 2 |
+---+---+
| 3 | 4 |
+---+---+---+---+
| 5 | 6 | 7 | 8 |
+---+---+---+---+
Count = 8 unit squares.
Area = 8 square units.

Counting Half Squares

Sometimes a shape cuts across unit squares diagonally, creating half squares:

       +---+
      /| 1 |
     / |   |
    +--+---+

Two half squares combine together to make 1 full unit square!
- 2 half squares = 1 whole square unit

- 4 half squares = 2 whole square units


Chapter Practice Exercises

  1. State the area in square units for each figure: a. A rectangle covered by 3 rows of 4 unit squares b. A square covered by 5 rows of 5 unit squares c. A shape composed of 14 unit squares with no gaps or overlaps
  2. A figure on a grid covers 6 whole unit squares and 4 half unit squares. What is the total area of the figure?
  3. Which unit would you use to measure the area of a postage stamp: square inches or square miles? What about the area of a soccer field?
  4. Look at the two shapes: Shape A covers 12 unit squares. Shape B covers 15 unit squares. Which shape covers more surface area, and by how many square units?
  5. True or False: If two shapes have different outlines, they cannot have the same area. Explain using an example of 12 unit squares arranged in different ways.

Solutions and Step-by-Step Answers

  1. Areas: a. 3 rows of 4 = 12 square units. b. 5 rows of 5 = 25 square units. c. Area = 14 square units.
  2. 6 whole squares + 4 half squares: 4 half squares = 2 whole squares. Total area = 6 + 2 = 8 square units.
  3. Postage stamp: Square inches (or square centimeters). Soccer field: Square meters (or square yards).
  4. Shape B covers more surface area than Shape A (15 > 12). It covers 15 - 12 = 3 more square units.
  5. False. Two shapes can have completely different shapes and outlines while sharing the exact same area! For example, 12 unit squares can be arranged into a long 1 x 12 rectangle, a 2 x 6 rectangle, a 3 x 4 rectangle, or an L-shape. All four figures have an area of exactly 12 square units.