In short

(a+b)^2 is the area of a square whose side is a+b. Slice that square once horizontally and once vertically — at the points where a ends and b begins — and you get four pieces: a small square of area a^2, another of area b^2, and two rectangles of area ab each. The identity (a+b)^2 = a^2 + 2ab + b^2 is just the equation "big square area = sum of four piece areas". Why: area is invariant under dissection — cutting a region into pieces and rearranging them never changes the total area.

You have probably written (a+b)^2 = a^2 + 2ab + b^2 a hundred times. You have probably also, at least once, written the wrong version — (a+b)^2 = a^2 + b^2 — on a homework problem and lost two marks. The trouble is that the identity, written as a string of symbols, looks arbitrary. Why does the cross-term 2ab exist? Where does the 2 come from?

The answer is so visual that once you see it, you cannot un-see it. And the picture is two and a half thousand years old — Euclid drew it in Elements, Book II, Proposition 4, where it appears not as algebra at all but as a theorem about literal squares and rectangles.

The picture

Draw a square. Make its side a+b long — that is, mark a point on each side that splits it into a piece of length a and a piece of length b. Now connect the marks across the square: one horizontal line and one vertical line. The big square is sliced into four rectangles.

Geometric proof of (a + b) squared as a big square cut into four piecesA large square of side a plus b. Top-left: a smaller square of side a, area a squared, shaded wheat. Top-right: a rectangle of width b and height a, area a b, shaded light blue. Bottom-left: a rectangle of width a and height b, area a b, shaded pale green. Bottom-right: a smaller square of side b, area b squared, shaded light yellow. Side lengths a and b are labelled along the top and left edges. a² ab ab b² a b a b
The big square has side $a+b$, so its area is $(a+b)^2$. The four pieces inside have areas $a^2$, $ab$, $ab$, and $b^2$. Same total area, two ways of writing it: $(a+b)^2 = a^2 + 2ab + b^2$.

That is the entire proof. Look at the top-right and bottom-left rectangles. Both have one side of length a and the other of length b, so each has area ab. Two rectangles, each of area ab, give you the 2ab in the middle of the identity. The 2 is not an algebraic coincidence — it is the count of a \times b rectangles in the picture.

Why: when you slice a region into non-overlapping pieces, the sum of the piece areas equals the original area. This is "area is invariant under dissection" — a basic geometric truth Euclid took as obvious. The identity (a+b)^2 = a^2 + 2ab + b^2 is just this dissection principle, written algebraically.

Worked examples

Take a = 4 and b = 3. Then a + b = 7, so the big square has side 7 and area 49. Now count the pieces:

  • the a \times a square has area 4 \times 4 = 16,
  • the b \times b square has area 3 \times 3 = 9,
  • each of the two a \times b rectangles has area 4 \times 3 = 12, so together they contribute 24.

Add: 16 + 9 + 24 = 49. The picture and the arithmetic agree.

If you had naively written (4+3)^2 = 4^2 + 3^2 = 16 + 9 = 25, you would have missed the two grey rectangles entirely — and missed 24 out of 49, almost half the square. The cross-term is not a small correction; in this example it is the largest of the three contributions.

Why (a+b)^2 \neq a^2 + b^2. This is the single most common mistake in school algebra, and the picture explains exactly what is going wrong. When a student writes a^2 + b^2 for (a+b)^2, they are imagining only the wheat-coloured square and the yellow square in the picture. They have forgotten to count the two side rectangles — the light-blue one in the top-right corner and the pale-green one in the bottom-left.

Those two rectangles are real area. They each have one side of length a and one of length b. They sum to 2ab. Drop them, and you have lost the part of the big square that lies between the two corner squares.

So the next time your hand is about to write (x + 5)^2 = x^2 + 25, pause and ask: where are my two ab rectangles? In this case a = x and b = 5, so the missing rectangles each have area 5x, and together 10x. The right answer is x^2 + 10x + 25.

Squaring numbers in your head. Suppose you want 103^2. Long multiplication of 103 \times 103 takes a minute and three lines of working. The dissection picture lets you do it in your head in about five seconds.

Write 103 = 100 + 3, so a = 100 and b = 3. Now mentally place the four pieces:

  • a^2 = 100^2 = 10{,}000,
  • b^2 = 3^2 = 9,
  • 2ab = 2 \times 100 \times 3 = 600.

Add: 10{,}000 + 600 + 9 = 10{,}609. So 103^2 = 10{,}609. Done.

The same trick works whenever one of your two pieces is a round number. 206^2 = (200 + 6)^2 = 40{,}000 + 2{,}400 + 36 = 42{,}436. 51^2 = (50 + 1)^2 = 2{,}500 + 100 + 1 = 2{,}601. Cricket scorers and shopkeepers across India use this kind of mental shortcut every day, often without ever having seen the picture that justifies it.

Where this comes from

Euclid wrote the Elements around 300 BCE. He had no algebra — no symbols a and b, no equals signs, no exponents. What he had was geometry. So the identity you know as (a+b)^2 = a^2 + 2ab + b^2 appears in Elements Book II, Proposition 4, in roughly these words:

"If a straight line be cut at random, the square on the whole is equal to the squares on the segments and twice the rectangle contained by the segments."

That is the dissection picture, in prose. "The square on the whole" is the big square. "The squares on the segments" are the wheat and yellow corner squares. "Twice the rectangle contained by the segments" is the two ab pieces. Book II of the Elements is sometimes called geometric algebra because almost every proposition is an algebraic identity disguised as a statement about areas.

The reason this matters today is that the picture survives the centuries because it is true at every scale and for every value. Plug in a = 4, b = 3, or a = 100, b = 3, or a = x + 1, b = y — the dissection still works, the four areas still add to the whole. Algebraic identities, at heart, are just clean ways of recording geometric truths that hold for every choice of side length.

Going deeper

Once the dissection picture clicks, a whole family of related identities opens up. The same kind of slice-and-count proof works for (a-b)^2 (a square with two strips removed and a corner added back), for (a+b+c)^2 (a big square cut by two horizontal and two vertical lines, giving nine pieces and the famous "three squares plus six rectangles" expansion), and for (a+b)^3 (a cube cut by three planes into eight rectangular boxes).

The general principle is called completing the square, and it shows up across mathematics — in solving quadratic equations, in deriving the formula for a circle from x^2 + y^2 = r^2, in finding maxima and minima of parabolas, and even in Gaussian integrals in statistics. Every time you "complete the square" in algebra, you are mentally adding back a missing b^2 corner to make the picture whole.

For a deeper look at how this style of geometric reasoning was extended by mathematicians like al-Khwarizmi in 9th-century Baghdad, who solved quadratic equations entirely with diagrams of squares and rectangles, see the references below.

References

  1. Wikipedia: Completing the square — historical and modern treatments of the same dissection idea.
  2. Wikipedia: Euclid's Elements, Book II — Proposition 4 is the original geometric statement of (a+b)^2.
  3. Khan Academy: Squaring binomials — short video walking through (a+b)^2 with a similar picture.
  4. NCERT Class 8 Mathematics, Chapter 9: Algebraic Expressions and Identities — the Indian school textbook section that introduces this identity, with a near-identical area diagram.
  5. Cut the Knot: Visual proofs of algebraic identities — a small gallery of dissection proofs of (a+b)^2, (a-b)^2, and a^2 - b^2.