Perimeter & unit conversion
Perimeter of 2-D shapes; convert between metric units (linked to place value/decimals); approximate metric ↔ imperial equivalences (inches, pounds, pints; miles ↔ km).
Perimeter is the distance all the way around the edge of a flat shape. This page also covers how to swap a length between measuring units — like centimetres and metres — and how to get a rough equivalent between metric and imperial units.
What it means
The perimeter of a 2-D shape is the total length of its boundary — imagine an ant walking right round the edge and back to where it started; the distance it walks is the perimeter. Because it is a length, you measure it in length units (mm, cm, m, km) and you find it by adding up the lengths of all the sides.
For a rectangle 6 cm wide and 4 cm tall, walk the four sides: 6 + 4 + 6 + 4 = 20 cm. A shortcut: opposite sides of a rectangle are equal, so perimeter = 2 × (width + height).
A unit is the size of one “step” you measure with. The metric system is built on tens, so converting is just moving a decimal point:
- 10 mm = 1 cm
- 100 cm = 1 m
- 1000 m = 1 km
Because 1 m = 100 cm, converting metres to centimetres means multiplying by 100 (2.5 m = 250 cm); going the other way you divide by 100 (70 cm = 0.70 m). This is exactly the place-value idea from N4.1: multiplying by 100 shifts every digit two columns to the left.
Imperial units (inches, pounds, pints, miles) don’t run on tens, so we only approximate between the two systems. Useful rough equivalences: 1 inch ≈ 2.5 cm, 1 mile ≈ 1.6 km, 1 kg ≈ 2.2 pounds, 1 litre ≈ 1.75 UK pints. The symbol ”≈” means “is roughly equal to”.
Worked examples
1. Perimeter of an L-shape. Add every side, even the short indented ones. An L-shape with edges of 4, 3, 2, 6, 5 and 1 cm going round has perimeter 4 + 3 + 2 + 6 + 5 + 1 = 21 cm. The trick is to keep your finger on the outline so you count each edge exactly once.
2. Same shape, different units. A square garden has sides of 3 m. Its perimeter is 4 × 3 = 12 m. In centimetres that is 12 × 100 = 1200 cm — same distance, smaller unit, so a bigger number.
3. Metric ↔ imperial estimate. A road sign says 80 km to the next town. Roughly how many miles? Divide by 1.6: 80 ÷ 1.6 = 50 miles.
| Given | Convert | Answer |
|---|---|---|
| 250 cm | to m | 2.5 m |
| 4 km | to m | 4000 m |
| 12 inches | to cm | ≈ 30 cm |
The generative-art connection
Perimeter is literally a path — so it draws itself. In a turtle tool like Turtle Blocks or Scratch’s pen, a program that walks the boundary of a shape is the perimeter: forward 100, turn 90 repeated four times both draws a square and traces its perimeter of 4 × 100 steps. Change the side length and the drawn outline grows in exact step with the number you are computing.
On the Geoboard, you stretch a rubber band around a lattice of pegs and count the unit segments along the band — the shape and its perimeter appear together, and nudging one peg lets you see the total change. This turns an abstract sum into something you build and watch, which is the whole mathartcademy premise: the art’s structure is the mathematics.
Common misconceptions
- Perimeter is not area. Perimeter is the distance around (a length, in cm); area is the space inside (in cm²). Two shapes can share a perimeter yet enclose very different areas.
- Forgetting hidden sides. On L-shapes and staircases, every edge counts — it is easy to miss the short indented ones.
- Converting the wrong way. To go to a smaller unit (m → cm) you multiply; to a larger unit (cm → m) you divide. If your number of centimetres came out smaller than your number of metres, you have flipped it.
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.
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A rectangle is `8 cm` long and `5 cm` tall. Find its perimeter, then check your answer a second way.
Answer
`8 + 5 + 8 + 5 = 26 cm`. Check with the shortcut: `2 × (8 + 5) = 2 × 13 = 26 cm`. Both agree, so the perimeter is `26 cm`.
Art hook Draw the rectangle on a Canvas grid, then animate a bright dot walking the outline one edge at a time; a running counter beside it adds each side length so the number `26` builds up exactly as the dot returns to its start.
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A rectangular garden bed is `2 m` long and `80 cm` wide. Its sides are given in different units. Find the perimeter in metres. (Hint: get both sides into the same unit first.)
Answer
Convert the width to metres: `80 cm = 80 ÷ 100 = 0.8 m`. Now add all four sides: `2 + 0.8 + 2 + 0.8 = 5.6 m` (or `2 × (2 + 0.8) = 2 × 2.8 = 5.6 m`). You must match units before adding, or the answer is meaningless.
Art hook On a Canvas grid where 1 square = 20 cm, draw the bed; a dot walks the outline while a readout shows the running total switching a `0.8 m` chip on for each short side, so the mixed-unit sum lands on `5.6 m`.
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Convert: (a) `3.5 m` into centimetres, (b) `450 cm` into metres, (c) `2 km` into metres.
Answer
(a) `3.5 × 100 = 350 cm`. (b) `450 ÷ 100 = 4.5 m`. (c) `2 × 1000 = 2000 m`. To a smaller unit you multiply; to a larger unit you divide.
Art hook A horizontal number line where a slider shows the same length in mm, cm, m and km at once; the digits slide left or right by the right number of places as you switch units, making the ×10 place-value shift visible.
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A square tile has a perimeter of `36 cm`. How long is one side? Explain how you know.
Answer
A square has 4 equal sides, so one side is `36 ÷ 4 = 9 cm`. (Working backwards: perimeter = side × 4, so side = perimeter ÷ 4.)
Art hook Show four equal squares of side `9` snapping together edge-to-edge into a larger square; a control lets you set the perimeter and the tool divides by 4 to draw the correct side length instantly.
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A running track is signposted as `5 km`. About how many miles is that? Use `1 mile ≈ 1.6 km`.
Answer
Divide by 1.6: `5 ÷ 1.6 ≈ 3.1 miles`. The `≈` means this is a rough equivalent, since metric and imperial units don't line up exactly.
Art hook Two parallel number lines, one in km and one in miles, sharing a draggable marker; as you drag, both readouts update using the ×1.6 factor so you see the same distance in two systems side by side.
Training exercises
Practice problems, easier first, that build toward the bar above. Each doubles as a seed for an interactive artwork.
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What does the word *perimeter* mean? Say it in your own words.
Answer
The perimeter is the total distance all the way around the edge of a flat shape — the length of its boundary. (Think of an ant walking right round the outline back to where it started.)
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A triangle has sides of `3 cm`, `4 cm` and `5 cm`. What is its perimeter?
Answer
Add the three sides: `3 + 4 + 5 = 12 cm`.
Art hook Place 3 dots on a Canvas as triangle corners; draw the edges and label each with its length, then sum them in a small badge at the centre so the total grows as each edge is drawn.
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A rectangle is `10 cm` wide and `6 cm` tall. Find its perimeter by adding all four sides.
Answer
`10 + 6 + 10 + 6 = 32 cm`. (Or `2 × (10 + 6) = 32 cm`.)
Art hook Animate a dot tracing the rectangle edge by edge; each side flashes its length as the dot passes, and a counter accumulates toward `32`.
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How many centimetres are in `1` metre? How many metres are in `1` kilometre?
Answer
`1 m = 100 cm` and `1 km = 1000 m`. The metric system is built on tens, hundreds and thousands.
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Convert `600 cm` into metres.
Answer
Going to a larger unit, so divide by 100: `600 ÷ 100 = 6 m`.
Art hook A stack of one hundred `1 cm` tiles collapses into a single `1 m` bar with an animation; repeat six times to show `600 cm` becoming `6 m`.
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A square has sides of `7 cm`. What is its perimeter? Try to use a multiplication shortcut.
Answer
All four sides are equal, so `4 × 7 = 28 cm`.
Art hook Rotate one `7 cm` segment through four quarter-turns to build a square; each placement adds `7` to a running total, landing on `28` as the square closes.
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An L-shaped patio has edges, going round, of `6 m`, `2 m`, `3 m`, `4 m`, `3 m` and `6 m`. Find its perimeter. Careful — count every edge once.
Answer
`6 + 2 + 3 + 4 + 3 + 6 = 24 m`. Keeping a finger on the outline helps you catch the short indented sides.
Art hook Draw the L on a Canvas grid and highlight each edge in a new colour as it is counted, greying it out afterwards so no edge is counted twice; the tally shows `24`.
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Two children measure the same fence. Sam writes `4 m = 40 cm`. Is Sam right? If not, fix it and say what went wrong.
Answer
Sam is wrong. `1 m = 100 cm`, so `4 m = 4 × 100 = 400 cm`. Sam multiplied by 10 instead of 100 (mixed up the mm↔cm step with the cm↔m step).
Art hook Show a `4 m` bar; when the wrong answer `40 cm` is chosen, only a tiny sliver lights up (clearly too short), and the correct `400 cm` fills the whole bar — the mistake becomes visually obvious.
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A rectangle has a perimeter of `20 cm`. Its width is `4 cm`. What is its height?
Answer
Two widths use `4 + 4 = 8 cm`, leaving `20 − 8 = 12 cm` for the two heights, so each height is `12 ÷ 2 = 6 cm`.
Art hook An interactive rectangle with a fixed perimeter budget of `20`: drag the width and the height auto-adjusts to keep the outline length constant, so you feel the trade-off between the two dimensions.
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A grown-up's big stride is about `36` inches. Roughly how many centimetres is that? Use `1 inch ≈ 2.5 cm`.
Answer
`36 × 2.5 = 90 cm`. So one big stride is roughly `90 cm` — a bit under a metre.
Art hook Animate a foot stepping out a `36`-inch stride against a metric ruler that fills in `2.5 cm` per inch, so the growing centimetre count reaches `90` as the stride completes.
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A regular hexagon (all sides equal) has a perimeter of `48 cm`. How long is one side?
Answer
A hexagon has 6 equal sides, so one side is `48 ÷ 6 = 8 cm`.
Art hook Place 6 dots evenly on a circle and join them into a hexagon; a control sets the perimeter and the tool divides by 6 to size each `8 cm` edge — increase the number of dots to see how side length shrinks as sides multiply.
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Which of these is the longest distance: `1500 m`, `2 km`, or `180000 cm`? Put them in order from shortest to longest.
Answer
Convert all to metres: `1500 m`; `2 km = 2000 m`; `180000 cm = 1800 m`. Order shortest→longest: `1500 m < 180000 cm (1800 m) < 2 km (2000 m)`.
Art hook Three bars on one shared metre-scale axis; each length is entered in its own unit but all render against the same axis, so their true lengths can be compared at a glance and sorted.