Why is the area of a triangle half the base times the height?

Because any triangle is exactly half of a rectangle with the same base and the same height. Take a copy of the triangle, turn it upside down, and the two fit together into that rectangle with nothing left over and no overlap. The rectangle's area is base times height, so the triangle's is half of it — and this works for a leaning triangle too, as long as the height is measured straight up from the base.

How do you find the area of a circle?

Multiply pi by the radius squared. For a circle of radius 5 cm the area is pi times 25, which is about 78.5 square centimetres. The commonest error is using the diameter instead of the radius: if a question gives you a diameter of 10 cm, halve it first. Squaring the wrong number quadruples the answer.

Can two shapes have the same perimeter but different areas?

Yes, and the difference can be enormous. A 6 by 6 square and a 1 by 11 rectangle both have a perimeter of 24, but the square's area is 36 and the rectangle's is 11. For a fixed perimeter the area is largest when the shape is as close to a circle as it is allowed to get, and smallest when it is long and thin.

What is a net of a solid?

A net is the flat shape you would cut out and fold up to make a solid — six squares in a cross makes a cube. Nets are the bridge between two and three dimensions, and the practical reason to draw one is that the surface area of a solid is just the total area of its net, so a three-dimensional problem becomes a set of ordinary flat area sums.

Shapes, Area and Perimeter

Q: Where does pi come from?

Pi is what you get when you divide any circle's circumference by its diameter. The remarkable part is that the answer is the same for every circle, large or small — roughly 3.14159. Because the ratio never changes, one number is enough to link the distance round a circle to the distance across it, which is why the same constant appears in both the circumference formula and the area formula.

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 distance all the way round the edge; area is how much surface is inside. A 5 cm by 3 cm rectangle has perimeter 5 + 3 + 5 + 3 = 16 cm and area 5 × 3 = 15 cm². Area and perimeter teaches them side by side deliberately, because the confusion is almost always between them rather than within either.

The units settle it every time. Perimeter is a length, so it is measured in centimetres or metres — you could walk it. Area is a covering, so it is measured in square centimetres, because you are counting how many 1 cm tiles fit. If an answer to an area question comes out in plain cm, something has gone wrong upstream.

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?

Because a triangle is exactly half of a rectangle with the same base and height. Take a second copy of the triangle, rotate it half a turn, and the two slot together into that rectangle with no gap and no overlap. Area of a triangle shows the fit, which is more convincing than any formula.

Base 8 cm, height 5 cm → 8 × 5 ÷ 2 = 20 cm².

"Height" means the perpendicular height — straight up from the base, not the length of a sloping side. A leaning triangle with the same base and the same perpendicular height has exactly the same area as an upright one, because sliding the top vertex sideways shears the shape without changing how much of it there is. That fact is why a sloping side is almost always a distractor in an exam question.

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 levelled 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?

Divide any circle's circumference by its diameter and you get the same number, about 3.14159, no matter how big the circle is. That constancy is the whole discovery, and circumference and diameter is built around measuring it rather than being told it.

A circle of diameter 10 cm has circumference π × 10 ≈ 31.4 cm.

Area of a circle gives πr², and the reason it involves the same 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 2πr, so πr. Multiply, and the area is πr².

The most expensive error in this topic is using the diameter where the formula wants the radius. Area depends on the radius squared, so halving the wrong number does not make you a bit wrong, it makes you four times wrong. When a question gives a diameter, the first line of working should be halving it.

How do you measure a solid?

Volume counts unit cubes the way area counts unit squares. Volume of a cuboid multiplies length by width by height, and the reason is the same layer-by-layer argument: the bottom layer holds length × width cubes, and there are height layers of them.

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 recognise 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, and the answer picks itself. Fencing a garden, edging a rug, or running skirting board round a room are perimeter jobs — you are buying a length. Turf, carpet, tiles and paint are area jobs — you are buying a covering. Sand for a sandpit or water for a tank is a volume job.

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.

It also explains why the units are worth caring about rather than tacking on at the end. A shop sells fencing by the metre and turf by the square metre, and those are different products at different prices. The unit in your answer is a statement about what kind of quantity you calculated, which is why an answer in cm to an area question is not a small slip — it means a length was computed where a covering was wanted.

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. 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.

Learn this properly in the app

Math Challenge teaches each of these as an illustrated lesson with a picture, a worked example and try-it problems, inside a catalog of 800+ lessons running from first counting up to calculus. 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.

Your turn

Three to try — tap what you get.

A 7 by 4 rectangle: what is its area?

The same 7 by 4 rectangle: its perimeter?

A triangle with base 10 and height 6: area?

Start with Math Challenge →