Picture a garden lawn. If you want to put a fence around it, you care about the perimeter — the total distance all the way round the edge. If you want to lay turf to cover it, you care about the area — the amount of flat space inside. Same lawn, two completely different measurements, asked for two completely different reasons.
And here is the thing that confuses people for years until someone says it plainly: they're measured in different units. Perimeter is a length, so it's in centimetres or metres — cm, plain. Area is a covering of squares, so it's in square centimetres or metres — cm², with the little raised two. That tiny symbol isn't decoration; it's telling you the measurement counts squares, not steps.
Area asks how many 1 cm × 1 cm squares fit inside a shape. Each square is one “square centimetre”. A 4 by 3 rectangle holds 4 × 3 = 12 of them, so its area is 12 cm². The little raised 2 is a memory of the two sides you multiplied together — length and width, both lengths, making something two-dimensional.
Two shapes with the same perimeter can have wildly different areas. A 1 cm by 8 cm strip and a 3 cm by 3-ish square can share a perimeter yet hold very different space — which is why, for a fixed length of fence, a square garden holds more lawn than a long thin one. Of all shapes, the circle squeezes the most area into a given perimeter, which is partly why bubbles are round.
Four straight-sided shapes cover almost everything you'll meet, and their formulas are all secretly the same idea: base times height, with a small adjustment. Learn the rectangle first, and the rest are just clever rearranging of it.
| Shape | Area formula | Why |
|---|---|---|
| Rectangle | base × height | count the rows of squares |
| Triangle | ½ × base × height | half of a rectangle |
| Parallelogram | base × height | a leaning rectangle |
| Trapezium | ½(a + b) × height | average of the two parallel sides |
For a triangle or parallelogram, “height” means the straight-up distance from base to top — the perpendicular height — not the length of a sloping side. A leaning parallelogram and an upright rectangle of the same base and height have exactly the same area, because sliding the top sideways doesn't change how much space is inside.
The word trapezium comes from the Greek for “little table” — a shape with a flat top and a flat bottom that aren't the same width, like a table seen end-on. Confusingly, what the British call a trapezium, Americans call a trapezoid, and what they call a trapezium, we'd call something else entirely. Same shape, two countries, swapped names.
A circle has just three measurements worth naming, and they all flow from a single point at the centre. The radius is the distance from the centre to the edge. The diameter is straight across through the centre — so it's always two radii, twice the radius. And the circumference is the whole distance around the edge: a circle's perimeter, by its proper name.
Now the famous one. If you measured the circumference of any circle — a coin, a wheel, a planet — and divided it by that circle's diameter, you'd always get the same number: about 3.14. Every time. Big circle, small circle, it never changes. That fixed number is π (“pi”). So π isn't really a mystery — it's just the answer to one simple question: how many diameters fit around the edge of a circle? The answer is a little over three, for every circle that has ever existed.
Because π is “circumference ÷ diameter”, turning it around gives the distance round: C = π × d (or π × 2r). The area of a circle is A = π × r² — “pi r squared”. Same π in both; the difference is that circumference uses the diameter once, while area uses the radius squared, because area is always a two-dimensional, squared kind of measure.
π never stops and never repeats. Computers have now worked it out to more than a hundred trillion digits, and there's still no pattern. Yet for almost anything you'll ever build, 3.14 is close enough; NASA uses only about fifteen digits to navigate spacecraft across the solar system to within centimetres.
You only need the two formulas from the last page. The whole skill is reading the question to see whether you've been given the radius or the diameter, then feeding the right one in. Take a circle with radius 5 cm — so its diameter is 10 cm — and use π ≈ 3.14.
πr² means π × (r × r), not (π × r) × 2. You square the radius first, then multiply by π. A common slip is to double instead of square, or to multiply by π before squaring. Work left to right in the right order: square the radius, then bring in π.
A pizza's area grows much faster than its width. An 18-inch pizza isn't twice a 9-inch one — because area depends on the radius squared, it's actually four times as much pizza. This is why the bigger size is nearly always the better value: you're paying for a number that's been squared.
Real shapes — an L-shaped room, a running track, a window — aren't tidy rectangles. But almost any shape can be chopped into ones you already know, worked out separately, and added back together. The whole method is: split, find each piece, add. Here is a worked L-shape.
Compound shapes often hide a side. If the bottom is 6 m and the top-left strip is 2 m wide, then the remaining top length must be 6 − 2 = 4 m. The lengths around any rectangle have to agree — opposite sides match — so a missing one can nearly always be found by adding or subtracting the sides you do know.
A 400-metre running track is a compound shape: two straights and two semicircular ends that join into one full circle. The reason the starting lines are staggered is pure geometry — a runner in an outer lane runs a bigger circumference round the bends, so they start further forward to make every lane exactly 400 m. The stagger is the π calculation, painted on grass.
Here is a circle with four pins. Pick a label below, then drop it on the matching pin. Two of the chips belong to other shapes entirely — read each one before you place it.
Fresh one. A rectangle is 8 cm long and 5 cm wide. Find its perimeter.
Fresh one. A triangle has base 8 cm and height 5 cm. Find its area.
Fresh one. A circle has a diameter of 20 cm. Find its circumference (π ≈ 3.14).
The 14th of March — written 3/14 the American way — is celebrated worldwide as Pi Day, and people eat pie to mark it. It also happens to be Albert Einstein's birthday. In 2015, at 9:26:53 on Pi Day, the date and time spelled out π to ten digits: 3.141592653.
A room is L-shaped. Splitting it gives two rectangles: one 5 m by 3 m and one 2 m by 2 m. You want to carpet the whole floor (so you need the area) and you also want to know roughly how the area compares with a plain 5 m by 4 m rectangular room. Work out the L-shaped room's total area by adding the two rectangles, then say which room would take more carpet, and explain in words why area is measured in m² and not in m. Show your steps.
Find each rectangle's area, add them for the L-shape. Work out the plain room's area too. Compare the two. Then one sentence on why the unit is squared.
strong You split the L-shape into the two rectangles and added their areas rather than guessing — 15 m² and 4 m² making 19 m². That “split, find each, add” method is exactly how compound shapes are marked, and you held the order.
try this State the comparison in one clear line — 19 m² against 20 m² — so the “which needs more carpet” answer is impossible to miss. A reader wants the verdict named, not just the two sums sitting nearby.
to add On the unit: say that area counts squares — how many 1 m by 1 m tiles cover the floor — so it has to be m², two lengths multiplied. Naming why the little two is there is the difference between getting the number and understanding it.
Two short films to watch alongside today's lesson — each shows you something the words and pictures can't.
You learned that perimeter is the walk round the edge and area is the space inside — and why one is in cm and the other in cm². You can find the area of a rectangle, triangle, parallelogram and trapezium, you know the parts of a circle and what π truly is, and you can chop a compound shape into pieces you already understand. Next time you measure a room or a wheel, Florence, you'll know exactly which question you're asking.
Around 1900, a man in Indiana nearly got a law passed that would have fixed π by decree — declaring it equal to a tidy 3.2 to make school sums easier. The bill sailed through the lower house before a visiting mathematician, in the building by chance, explained that you cannot legislate a number into being neat. π stayed exactly as endless as it always was.