ACARA v9 CONTENT DESCRIPTION “recognise the area of a shape as the number of square units that cover it and the perimeter as the distance around it, and measure both using grids and informal units”
A flat shape can be measured in two quite different ways, and the first is area: the amount of surface inside it. Area is counted in equal unit squares, so a shape that four unit squares cover has an area of four square units. On a grid this is just counting the squares that fill the shape. For a rectangle, the squares line up in equal rows, so counting becomes multiplying: rows times columns. Area answers how much surface a shape covers, and unit squares are the natural way to measure it, just as unit lengths measure distance.
Count the squares
Area is how many equal unit squares cover a shape.
This rectangle is 3 units by 2. How many unit squares cover it?
Perimeter is distance around
The second measurement is perimeter: the distance all the way around the edge of a shape. Walking the boundary one unit at a time and counting the steps gives the perimeter, and for a rectangle it is the four sides added together. A four by three rectangle has a perimeter of four plus three plus four plus three, which is fourteen units. Perimeter is a length, measured in units, not squares, because it follows the edge rather than filling the inside. It answers a different question from area: how far is it around, rather than how much is inside.
Walk the perimeter
Perimeter is the total distance around the edge of a shape.
Perimeter is the distance all the way around. Walk one unit edge at a time.
Two measurements, one shape
Area and perimeter are easy to confuse, but they measure different things about the same shape. Area is the surface inside, counted in square units; perimeter is the boundary around, counted in lengths. One shape has both at once, and the two are found in different ways: area by counting or multiplying the squares that fill it, perimeter by adding the lengths around its edge. Keeping them apart — inside versus around, squares versus lengths — is the central idea of this unit, because the same rectangle gives two numbers that mean entirely different things.
Area or perimeter?
Area measures the inside in squares; perimeter measures the edge in lengths.
The same rectangle has both an area and a perimeter. They measure different things.
Building and measuring
Choosing the sides of a rectangle fixes both its area and its perimeter at once, and watching them change makes the two formulas concrete. Make it wider and both grow, but not in step: area is the two sides multiplied, while perimeter is the two sides added and doubled. A three by two rectangle has area six and perimeter ten; stretch it and the numbers part ways. Building rectangles and reading off both measurements shows that area depends on the sides multiplied, perimeter on the sides added — the same two lengths used two different ways.
Build a rectangle
Choose the sides and watch both area and perimeter change.
A 3 by 2 rectangle has area 3 times 2 = 6 square units, and perimeter 10 units. Area uses both sides multiplied; perimeter adds them around.
Same area, different shape
A surprising and important fact is that shapes with the same area can have different perimeters. Twelve unit squares can be laid out as a long twelve by one strip, a six by two block or a four by three rectangle — all area twelve, but with perimeters of twenty-six, sixteen and fourteen. The long thin shape has the largest perimeter because more of its edge is exposed. This shows area and perimeter are truly independent: knowing one does not fix the other. It is why a farmer with a fixed amount of fence can enclose different areas depending on the shape, a real use of this idea.
Same area, different perimeter
Shapes with equal area can have different perimeters.
Every one of these has area 12 square units, but this 12 by 1 shape has perimeter 26. Same area can come with different perimeters.
Working out both
With area as squares inside and perimeter as distance around, both can be worked out for any rectangle: area by multiplying the two sides, perimeter by adding all four. A two by three rectangle has area six and perimeter ten; a four by four has area sixteen and perimeter sixteen, a reminder that the two being equal in number is a coincidence, not a rule. Reading a table of rectangles makes the two methods routine and keeps them separate. With area and perimeter measured on grids, told apart, and seen to vary independently, a child has the foundation for the formal area and perimeter formulas, and for measuring real surfaces and boundaries in the years ahead.
Area and perimeter table
Each rectangle has its own area (sides multiplied) and perimeter (sides added).
For each rectangle, area multiplies the sides and perimeter adds them around. Reveal each row.
Quick self-check
1. The perimeter of a rectangle is...
2. On a unit grid, the area of a shape is...
3. Area and perimeter are...
4. A rectangle 3 units by 4 units has an area of...
5. Two rectangles both have area 12 square units. Their perimeters...