Area
Area (count squares → area of rectangles = l × w); area units (cm², m²); convert area units.
Area is the amount of flat space a shape covers — how much surface is inside its outline.
What it means
Length measures how far along something is; area measures how much surface a flat shape takes up. To measure it, we cover the shape with identical squares and count them.
Pick a square as your unit — say a square 1 cm on every side, called a square centimetre and written cm². Lay these unit squares edge to edge over the shape with no gaps and no overlaps. The number of squares it takes is the area. Cover a shape with 12 such squares and its area is 12 cm². That is all “square centimetre” means: the space of one 1-cm-by-1-cm tile.
For a rectangle you do not have to count one square at a time. If it is 5 cm long and 3 cm wide, the squares line up in a neat grid: 3 rows with 5 squares in each. That is a 5 × 3 array, so 15 squares. This gives the rule
area of a rectangle = length × width
The multiplication is the counting — rows times columns — which is why you needed your times-tables first. A square is just a rectangle with equal sides, so a 4-cm square has area 4 × 4 = 16 cm².
Units name the size of the tile. cm² uses a 1-cm tile; m² (square metre) uses a 1-metre tile. Because 1 m = 100 cm, one square metre is a 100-by-100 grid of centimetre squares, so 1 m² = 100 × 100 = 10 000 cm². Converting area is not the same as converting length — you scale by the factor twice, once for each direction.
Worked examples
- Count the squares. A shape drawn on grid paper covers 7 whole squares plus two half-squares. Area =
7 + 1 = 8squares. - Use the rule. A rug is 6 m long and 4 m wide. Area =
6 × 4 = 24 m². - A square. A tile is 30 cm on each side. Area =
30 × 30 = 900 cm². - Convert units. A table top is
2 m². In square centimetres that is2 × 10 000 = 20 000 cm².
The generative-art connection
An array of dots is a multiplication, and a filled rectangle is its area. In dot-multiplier, one dot multiplies into rows and columns of dots; a 5 × 3 block of dots is exactly the 15 unit squares of a 5-by-3 rectangle, made visible. Change the width and the whole area grows or shrinks in front of you — you watch length × width happen.
On the Geoboard, stretch a rubber band into any rectangle and the enclosed grid squares are the area you can literally count. Nudge one corner and the number changes, so the formula stops being a rule to memorise and becomes something you see. The same move — tile a region, count the tiles — is the seed of tessellation art, where repeated shapes cover a plane with no gaps.
Common misconceptions
- Mixing up area and perimeter. Perimeter is the distance around the edge (add the sides); area is the space inside (count the squares). Two shapes can share a perimeter yet have very different areas.
- Adding instead of multiplying. For a 5-by-3 rectangle the area is
5 × 3 = 15, not5 + 3 = 8. Picture the rows of squares to see why. - Converting area like length.
1 mis100 cm, but1 m²is10 000 cm², not100. You are covering both directions, so multiply by the factor twice.
What a learner should be able to solve
The benchmark for mastering this concept — the problems a child at this stage is expected to be able to work through.
-
A garden bed is `8 m` long and `5 m` wide. What is its area? Remember what "area" means and pick the right operation.
Answer
Area of a rectangle = length × width, so `8 × 5 = 40 m²`. (Not `8 + 5` — that would be perimeter thinking.)
Art hook Draw an 8×5 grid of unit-square tiles on Canvas; fill them one row at a time on a timer so the child watches 40 squares appear, with a running count label. Sliders for length and width regrow the grid live.
-
A square floor tile measures `20 cm` on each side. What is the area of one tile in `cm²`?
Answer
A square is a rectangle with equal sides, so area = `20 × 20 = 400 cm²`.
Art hook Tile a canvas with squares; clicking one highlights its 20×20 = 400 unit-square interior as a mini-grid, so the big tile and its area appear together.
-
On grid paper a leaf shape covers `11` whole squares and `6` half-squares. What is its area in squares?
Answer
Pair the halves: `6` halves make `3` whole squares. Area = `11 + 3 = 14` squares.
Art hook A pixel-grid painter where whole cells count as 1 and half-shaded cells (a triangle fill) count as 0.5; a live tally sums the area as you paint a shape.
-
A rectangular tabletop is `2 m` long and `3 m` wide. First find its area in `m²`, then convert it to `cm²`. (Hint: `1 m² = 10 000 cm²`.)
Answer
Two steps. Area = `2 × 3 = 6 m²`. Convert: `6 × 10 000 = 60 000 cm²`. You scale by 100 twice — once for each direction — not just once.
Art hook Zoom animation: build the 2×3 = 6 m² rectangle from big square-metre blocks, then 'zoom in' on one block to reveal its 100×100 array of tiny cm² dots, so the ×10 000 is visible before scaling up to all 6 blocks (60 000).
-
A rectangle is `7 cm` long and covers an area of `28 cm²`. How wide is it?
Answer
Area = length × width, so width = `28 ÷ 7 = 4 cm` (division is the inverse of the area rule). Check: `7 × 4 = 28 cm²`.
Art hook Fix the area at 28 unit squares and let the user drag one side; the other side auto-adjusts to keep 28 squares, showing all the rectangles (7×4, 14×2, 28×1) that share the same area.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
-
What does the word "area" tell you about a shape — the distance around its edge, or the amount of flat space inside it?
Answer
The amount of flat space inside it (the surface it covers). The distance around the edge is called perimeter.
Art hook Two buttons over one shape: 'trace edge' lights up the outline (perimeter); 'fill inside' floods the interior with colour (area) — a one-tap way to feel the difference.
-
A shape on grid paper covers exactly `6` whole unit squares with no bits left over. What is its area?
Answer
`6` squares. Area is just the number of unit squares that cover the shape.
Art hook Click cells on a blank grid to build any shape; a counter shows the area growing by 1 with each square you add.
-
A rectangle is `4` squares long and `3` squares wide. Count the rows and columns — how many unit squares cover it? Which is quicker: adding or multiplying?
Answer
`4 × 3 = 12` squares. Multiplying (rows × columns) is quicker than counting one by one; adding the two side-lengths (`4 + 3 = 7`) gives the wrong answer.
Art hook dot-multiplier style: a single dot fans out into a 4×3 array of dots; toggle between 'count each' and 'multiply' to see 12 either way.
-
Use the rule `area = length × width`. A poster is `9 cm` long and `6 cm` wide. What is its area?
Answer
`9 × 6 = 54 cm²`.
Art hook Slider-driven rectangle on Canvas that prints length × width = area live; the fill grows so the number always matches the coloured region.
-
A square patch of grass is `7 m` on every side. What is its area in `m²`?
Answer
`7 × 7 = 49 m²` (a square is a rectangle with all sides equal).
Art hook Rotating hue on a square whose side you set; the area label pulses so a 7-side square glows brighter/bigger than a 3-side one — area made felt.
-
Spot the mistake. Sam says: "My rectangle is `5 cm` by `2 cm`, so its area is `5 + 2 = 7 cm²`." What did Sam do wrong, and what is the correct area?
Answer
Sam added the sides instead of multiplying. Area = `5 × 2 = 10 cm²`. (Adding the sides is on the way to perimeter, not area.)
Art hook Split-screen: left tiles the true 5×2 = 10 squares; right shows Sam's '7' as a broken, gap-filled attempt that visibly can't cover the rectangle.
-
A book cover is `2` squares long and covers `10` squares in total on the grid. Its width was smudged — how many squares wide is it?
Answer
Width = `10 ÷ 2 = 5` squares (undo the area rule with division). Check: `2 × 5 = 10`.
Art hook A 'reveal the missing side' game: the area (10 squares) is shown filled, one side is hidden behind a curtain, and dragging a slider to width 5 makes the grid snap to exactly fill.
-
A hallway is `12 m` long and `2 m` wide. What is its area? What is its perimeter? Notice they are different measures.
Answer
Area = `12 × 2 = 24 m²`. Perimeter = `12 + 2 + 12 + 2 = 28 m`. Area counts inside squares; perimeter measures the edge — different jobs, different answers.
Art hook One rectangle with two overlaid readouts: a glowing outline animating around the border (28 m) and a fill sweeping the interior (24 squares) at the same time.
-
Find TWO different rectangles (whole-number sides) that each have an area of exactly `12` squares. Give the length and width of each.
Answer
Any two of: `1 × 12`, `2 × 6`, `3 × 4` (or their flips). Each multiplies to `12`.
Art hook A factor-rectangle explorer: type an area (12) and the canvas draws every whole-number rectangle with that area side by side, revealing the factor pairs as shapes.
-
Convert. A tabletop is `3 m²`. Using `1 m² = 10 000 cm²`, how many `cm²` does it cover?
Answer
`3 × 10 000 = 30 000 cm²`. Area converts by scaling twice (×100 for length, ×100 for width = ×10 000).
Art hook A grid-of-grids: three big m² squares, each of which zooms to reveal 10 000 cm² dots, with a total counter climbing to 30 000.
-
Reasoning. Two rectangles both have a perimeter of `12 cm`. One is `5 cm` by `1 cm`; the other is `3 cm` by `3 cm`. Do they have the same area? Work out both.
Answer
No. First area = `5 × 1 = 5 cm²`; second = `3 × 3 = 9 cm²`. Same perimeter, different area — the more square-like shape holds more space.
Art hook Morph animation holding perimeter fixed at 12: a rectangle slides from 5×1 to 3×3 while its filled area count visibly swells from 5 to 9, peaking at the square.
-
A wall is `4 m` long and `3 m` high. It has one square window `1 m` by `1 m`. What area of wall is left to paint (not counting the window)?
Answer
Wall area = `4 × 3 = 12 m²`; window = `1 × 1 = 1 m²`. Paint area = `12 − 1 = 11 m²`.
Art hook Canvas wall of 12 unit squares you fill with 'paint' colour; click the window square and it stays blank, so the painted count settles at 11 — area by subtracting a hole.