Area — where it appears
Named by 39 essays across 7 fields — each of them below, with the objects they name alongside it.
Two squares, four triangles, and no algebra
The Pythagorean theorem is usually met as a formula to be memorised. It is much better met as a rearrangement that can be checked by eye.
Adding up rectangles until they stop being rectangles
The integral is defined as a limit of sums of rectangles. The definition is exact, the picture is honest about what it costs, and the gap between them is the whole subject.
A matrix is a picture of what happens to the grid
Four numbers in a box is not an object anyone has intuitions about. The same four numbers, shown as an instruction for redrawing the plane, are.
A circle unrolled into a triangle
Take a disc apart into rings, straighten each one, and stack them. The result is a triangle whose base is the circumference and whose height is the radius — and its area is the disc's.
Round is not the only way to be the same width
A shape that measures the same in every direction sounds like a description of a circle. It is not — there are infinitely many others, one of them is on a coin in most people's pockets, and a drill built from one cuts a nearly square hole.
Bayes' theorem is a picture of a square
A test that is 99% accurate returns a positive result. The chance it is right can easily be under one in five, and the reason is visible the moment the population is drawn as a square rather than described as a formula.
Euclid proves it without moving anything
The rearrangement proof cuts and slides. Euclid's does neither — it shows that a square and a rectangle are each exactly twice the same triangle, seen from opposite sides, and that is harder to hold in the head for a reason worth understanding.
Area is the undoing of slope
Two operations invented for unrelated reasons — measuring a region and measuring a rate — turn out to be inverse. The picture is two panels sharing one axis, and the claim is that the lower curve's steepness is the upper curve's height.
Completing the square, by completing a square
The step everybody is taught as an algebraic trick is a literal instruction about a literal square. There is a corner missing, its size is forced, and paying for it is the whole method.
The door that was not opened
Three doors, one prize, a host who opens a losing door and offers a swap. Switching wins two times in three, and the reason is not about doors — it is about what the host was allowed to do.
The number that says how much room is left
A linear map takes the unit square to a parallelogram. The area of that parallelogram is one number, it is computable from the four entries of the matrix, and almost everything the determinant is used for is a restatement of that sentence.
Area by counting dots
Draw a polygon with every corner on a grid of dots. Count the dots strictly inside, add half the dots on the edge, subtract one — and the answer is the area, exactly, with no measuring anywhere.
The most area a fence can hold
One length of boundary, and the question of what shape to bend it into. The answer is a circle, everybody knows it, and the argument that convinced the nineteenth century turned out to prove something slightly different.
A rectangle grown on two sides
A product of two changing quantities is the area of a rectangle whose sides both move. The extra area is two strips and a corner, and the whole of the product rule is the observation that the corner is negligible and the strips are not.
Equal area is enough, and equal volume is not
Any two polygons of the same area can be cut into each other with finitely many straight cuts. The same sentence with area replaced by volume and polygon by polyhedron is false, and what blocks it is an angle.
The area that names the number
The number e can be defined without mentioning slopes at all. Slide right along the curve 1/x until the area underneath reaches exactly one, and stop. That is where e is, and the reason logarithms turn multiplication into addition is visible in the same picture.
One point in every big enough shape
A determinant measures a lattice, not the basis that happened to describe it — and that measurement is an exchange rate. Any symmetric convex region with more than four times that area has to swallow a lattice point.
Angles survive and areas do not
Stereographic projection takes every circle on the sphere to a circle or a line, and every crossing angle to itself. It does both exactly, with no approximation anywhere, and it destroys area so thoroughly that a patch near the pole can be a thousand times its neighbour's size.
The least area a width can hold
Barbier's theorem says every curve of constant width has the same perimeter, which removes perimeter as a way of telling the family apart. Area is not like that — the circle holds the most and the Reuleaux triangle the least — and the reason the minimiser has corners is a constraint rather than a preference.
A dissection that never comes apart
The plane theorem lets the pieces be picked up and put down anywhere. Require instead that they stay joined at their corners and swing, and the theorem survives — which was open for a century and is a much stronger statement about the same cuts.
Slid, but never turned
The classical dissections all turn their pieces. Forbid the turn — allow the pieces to be slid and nothing else — and equal area stops being enough, for a reason that is a single number attached to each direction and that a cut cannot change.
Finitely many, and nobody says how many
The theorem promises a dissection exists and the proof produces one. Running the proof on a hexagon produces thirty-nine pieces, ingenuity produces five, and there is no method for proving that five cannot be four.
Equal area on a sphere, without a rectangle
On a sphere, two polygons of the same area can still be cut into each other, exactly as in the plane. Almost nothing in the plane proof survives the move: a sphere has no rectangles, no parallel strips and no similar triangles of different sizes. What carries the theorem instead is a quadrilateral with two right angles, built from a triangle's midline.
Half a circle against a wall
Lay a fence of fixed length with both ends against a straight wall and the best shape is a half-circle, holding exactly twice what a full circle of the same fence holds. The proof is a mirror: doubled in the wall, any fence becomes a closed curve with twice the length and twice the area, and the closed-curve answer carries over. In a corner the same mirrors give a slice of a circle — until the corner's angle stops dividing a half-turn.
The slice that has to match
The same slicing one dimension up gives the sphere's volume in a line, once one comparison is noticed: at every height a hemisphere's disc has exactly the area of a cylinder's slice with a cone's taken out of it. The principle that licenses that comparison also returns a false answer the moment the slices are not parallel.
The angle that is really an area
On a unit circle the angle t is the length of arc the point has walked, and it is also twice the area of the slice it has swept. The two readings agree on the circle and part company on the hyperbola — where arc length leads nowhere and area leads straight to cosh, sinh, and the exponential.
A rectangle cut by a curve
Integration by parts is taught as the product rule run backwards. It is also a picture: an increasing curve cuts a rectangle into two pieces, one of them the area under the curve and the other the area beside it, and the formula says only that the pieces fill the rectangle. Run repeatedly, the same cut produces the factorials and Wallis's product for π.
Two children and the sentence about one of them
A family has two children and at least one is a boy. The chance that both are boys is one in three — or one in two, or anything from one in three to certainty — and every one of those answers is right for some way the sentence could have come to be said. There is no host and no door, and the protocol is still the whole problem.
One coin, counted by runs and by wakings
Beauty is put to sleep and a fair coin is tossed. Heads, she is woken once; tails, twice, with the first waking erased from her memory. Each time she wakes she is asked how likely heads is. One half, say some; one third, say others; and unlike every earlier puzzle of this kind, stating the protocol exactly does not end the argument.
The least wall for equal rooms
Divide the plane into rooms of equal area using as little wall as possible, and every wall does double duty. Bees settled on hexagons long ago, and the proof that nothing does better — not even rooms with curved walls — came in 1999. The straight-walled half of it is two facts: the rooms of any division average six sides, and more sides never cost more wall.
Every chord slid to the middle
Take a shape, pick a line, and slide every chord that crosses the line at right angles until the line cuts it in half. The area cannot change, the boundary can only get shorter, and the result is symmetric. Do it again about another line, and another, and the shape is squeezed towards a disc — unless the lines are badly chosen, in which case it stops short.
No odd number of equal triangles
A square can be cut into two triangles of equal area, or four, or any even number. It cannot be cut into three, or five, or any odd number — whatever shapes the triangles take. Paul Monsky's proof of 1970 has no geometry in its engine at all: it colours the points of the plane by how divisible their coordinates are by two.
The count a fold cannot change
A curved map can fold the plane over itself, so that one point has three preimages and its neighbour has one. Count each preimage with the sign of the determinant there and the jump disappears — the signed count is the same everywhere, and it is a whole number.
A band of a sphere is a band of its cylinder
Cut a sphere with two parallel planes and the band between them has exactly the area of the band the same planes cut from the cylinder wrapped round the sphere — whatever the band's latitude. Near the equator the sphere's band is wide and nearly upright; near a pole it is narrow and nearly flat; and the two effects cancel exactly. Archimedes proved it, wanted it on his tomb, and it gives the area of the sphere, the only honest way to pick a random point on it, and a map on which no country is the wrong size.
An area measured by walking round it
A surveyor's instrument from 1854 measures the area of any shape on a map by having its pointer steered once round the boundary; a small wheel rolls and slides, and its reading, times the length of one arm, is the area. Nothing touches the inside. The reason is that area can be written as an integral over the boundary — Green's theorem — and the instrument is that integral built in brass. Walk a curve that crosses itself and the same integral counts some regions twice.
How nearly equal an odd number of triangles can be
A square cannot be cut into an odd number of triangles of equal area. It can be cut into five triangles whose areas differ by about two hundredths, seven that differ by three thousandths, nine by a ten-thousandth and a half — the spread falling by a factor of seven or more with every two triangles added, closing on equality and never reaching it. A search for the closest finds that the best five it can make have the golden ratio in their areas.
The area of the Mandelbrot set, from two sides
The Mandelbrot set has an area, and nobody knows it. Counting grid cells closes in on 1.5066 from both sides, but every count judges a cell by a single point, and points on the boundary never declare themselves. An exact formula exists — π times one minus a sum over the coefficients of the map onto the set's outside — and every partial sum is a guaranteed upper bound; after sixteen thousand coefficients the bound is still 1.786, and at the rate it falls it would need something like 10²⁵ more.
The five crescents that can be squared
The circle cannot be squared, but some regions bounded by circular arcs can: Hippocrates of Chios found a crescent with exactly the area of a triangle around 440 BC. The trick is to make two multiples of π cancel, and it works for a crescent whose two arcs turn through angles in the ratio m to n exactly when an angle defined by a polynomial can be drawn with ruler and compass. That happens for five ratios and no others.
The largest sofa round a corner
A hallway one unit wide turns a right angle. What is the largest shape that can be carried round? A half-disc gets to 1.571. John Hammersley cut a notch and, by one line of calculus, reached π/2 + 2/π = 2.2074. Joseph Gerver reached 2.2195 in 1992 with a boundary of eighteen analytic pieces and a value with no closed form. It took another thirty years for a proof that this was the end, and the problem began as a question about moving furniture.
Named alongside it
The objects these essays reach for when they reach for this one.
DissectionInvariantConvexityCounting argumentCircleCongruenceDeterminantFundamental theoremLimitOperation setPiPolygon