Shapes, Area and Perimeter

Updated 4 October 2026

Almost every area formula in school mathematics comes from one shape: the rectangle. A rectangle's area is length times width because that is literally how many unit squares fit inside it, and every other formula on this page is a way of turning some other shape into a rectangle you already know how to measure.

That is worth knowing before the formulas arrive, because a list of five formulas is hard to hold and one idea with five consequences is not.

What is the difference between area and perimeter?

Perimeter is the total distance around the edge of a shape, and area is the amount of flat surface inside it. For a rectangle 5 cm long and 3 cm wide, the perimeter adds all four sides: 5 + 3 + 5 + 3 = 16 cm. The area counts the centimeter squares that cover it, 5 in each of 3 rows, so 5 × 3 = 15 cm². Area and perimeter teaches the two side by side, because most mistakes come from mixing them up. Perimeter is a length, so it is measured in units such as cm or m. Area counts squares, so it is measured in square units, cm² or $m^2$m². For a shape made of rectangles, such as an L-shape, find every missing side before adding up the perimeter, and find the area by cutting the shape into rectangles and adding their areas. The units give a quick check: a result in cm² must be an area.

They are also genuinely independent. A 6 by 6 square and a 1 by 11 rectangle both have perimeter 24, but their areas are 36 and 11 — the same fence around wildly different amounts of field.

Why is a triangle's area half the base times the height?

A triangle's area is half the base times the height because two copies of it make a parallelogram, which has the same area as a rectangle with that base and height. Rotate the copy half a turn and the two slot together with no gap. Area of a triangle shows the fit. So a triangle with a base of 10 cm and a height of 7 cm has area 10 × 7 ÷ 2 = 35 cm². The height is the perpendicular distance from the base to the opposite corner, not the length of a sloping side. In a triangle with an obtuse angle, the height can fall outside the triangle, and it is drawn to the base extended with a dotted line. Any side can be the base, as long as the height is measured at a right angle to that side. The triangle's area must be less than the rectangle drawn around it: 35 is half of 70.

Base 8 cm, height 5 cm $\to 8\times 5÷2=20$→ 8 × 5 ÷ 2 = 20 cm².

The same dissection argument does the other two shapes. Area of a parallelogram is base times height because cutting a triangle off one end and moving it to the other turns the parallelogram into a rectangle. Area of a trapezium averages the two parallel sides and multiplies by the height, which is the rectangle you would get if you leveled the sloping top out — its width being the average of what it was at the bottom and at the top.

Where does pi come from?

Pi is the number you get when you divide any circle's circumference by its diameter. The answer is always the same, about 3.14159, however big or small the circle is. The lesson on circumference and diameter measures this rather than stating it: wrap a string around a can, compare it with the width, and the string is always a little more than three widths long. The ratio never changes because all circles are the same shape. Doubling the width doubles the circumference. That fact becomes a formula: circumference $=\pi \times$= π × diameter. Using the fraction $\frac{22}{7}$22/7, which is close to pi, a circle with a diameter of 7 cm has a circumference of about $\frac{22}{7}\times 7=22$22/7 × 7 = 22 cm. When a question gives the radius, double it first to get the diameter. A radius of 7 cm gives a circumference of about 44 cm. The circumference should be a bit more than three times the diameter: 3 × 14 = 42.

A circle of diameter 10 cm has circumference $\pi \times 10\approx 31.4$π × 10 ≈ 31.4 cm.

Area of a circle gives $\pi r^2$πr², and the reason it involves the same $\pi$π is worth seeing once. Cut the circle into many thin wedges and lay them alternately point-up and point-down: they interlock into something that gets closer and closer to a rectangle as the wedges get thinner. That rectangle's height is the radius, and its width is half the circumference — half of $2\pi r$2πr, so $\pi r$πr. Multiply, and the area is $\pi r^2$πr².

How do you measure a solid?

A solid is measured by its volume, the number of unit cubes that fill it, just as area counts unit squares. Volume of a cuboid multiplies length by width by height. A box 5 cm long, 4 cm wide and 3 cm high has volume 5 × 4 × 3 = 60 cm³. The formula works layer by layer. The bottom layer holds 5 × 4 = 20 centimeter cubes, and the box is 3 layers high, so it holds 3 × 20 = 60 cubes. Any prism works the same way: its volume is the area of its cross-section times its length. All three measurements must be in the same unit before multiplying, so a box 2 m long, 50 cm wide and 40 cm high is first written as 200 cm by 50 cm by 40 cm. A volume always has a cubed unit, such as cm³ or $m^3$m³, which makes a quick check.

Surfaces are handled by flattening. Nets unfolds a solid into the flat shape you would cut out and fold up — six squares in a cross for a cube, a rectangle and two circles for a cylinder — and once it is flat, the surface area is an ordinary sum of flat areas. The net is not a diagram for its own sake; it is the technique that turns a three-dimensional problem into one you can already do.

Naming shapes, and why it comes first

Before any of the measuring, a learner has to be able to recognize a shape reliably. Circles, triangles, squares and rectangles establishes the four flat shapes everything else is built from, and cubes, spheres, cones and cylinders does the same for solids.

The subtle lesson is that a shape stays itself when it moves. Shapes turned and resized exists because a triangle balanced on a vertex is still a triangle and a very long thin rectangle is still a rectangle — and young learners frequently believe otherwise, having only ever been shown one orientation.

Above, below and beside covers the language of position, which sounds like vocabulary and is actually the beginning of coordinates: describing where something is relative to something else is what a coordinate system formalises later.

Building shapes out of other shapes

Counting is where geometry meets number. Counting sides and corners gives a shape a description that does not depend on how it looks — four sides and four corners is what makes a shape a quadrilateral, however squashed.

Making shapes from smaller shapes is the dissection habit in its first form: two triangles make a square, six triangles make a hexagon. The same instinct — cut the awkward shape into ones you can measure — is what solves composite area questions years later.

Continuing a shape pattern looks like the lightest item here and is doing real work: spotting a repeating unit and predicting what comes next is the same skill that later reads a sequence and writes its nth term.

Angles and symmetry

An angle measures turn, not length — which is why a small angle drawn with very long arms is still a small angle. Angles introduces the measure with right angles as the anchor, and that anchoring matters: most angle estimation is done by comparison with 90°, not by imagining 37 individual degrees.

Symmetry asks where a shape could be folded so the halves match exactly. It is the first idea on this page that is about a shape as a whole rather than about a measurement of it, and it is the doorway into transformations — a line of symmetry is precisely a mirror that leaves the shape looking unchanged.

Which one does a real problem want?

Ask what is being bought or measured, and the answer picks the measurement. Fencing a garden, edging a rug, or fitting baseboard around a room is a perimeter job, because you are buying a length. Turf, carpet, tiles and paint are area jobs, because you are buying a covering. Sand for a sandbox or water for a tank is a volume job, because you are filling a space. Take a garden 8 m long and 5 m wide. The fence around it is 8 + 5 + 8 + 5 = 26 m, the turf to cover it is $8\times 5=40m^2$8 × 5 = 40 m², and soil spread $\frac{1}{10}m$1/10 m deep fills $40\times \frac{1}{10}=4m^3$40 × 1/10 = 4 m³. This works because each job is sold in its own unit. A fence with a 1 m gate needs only 26 − 1 = 25 m of fencing. Read the unit on the price list back against the answer before buying.

That framing is more reliable than remembering formulas, because it survives questions that never use the words. "How much ribbon goes round the edge of this cake" is a perimeter question that does not say so, and reading it as one is the whole of the difficulty.

The mistakes worth naming

• Answering area when the question said perimeter. Read the units in the answer line; they usually give the game away.
• Using a sloping side as the height. The height is always measured perpendicular to the base.
• Squaring the diameter. $\pi r^2$πr² wants the radius. Halve first, then square.
• Forgetting to halve for a triangle. Base times height is the rectangle; the triangle is half of it.

The geometry thread continues in lines, angles and Pythagoras, which turns the angle work here into proper reasoning, and polygons, solids and surface area, which takes nets much further.