The difference of squares, rearranged by hand.
a² − b² is not a square: it is a rectangle in disguise. Remove a corner b × b from a square of side a, cut what is left at a single point and move the piece across: you get an (a+b) × (a−b) rectangle with the same area. Here you can watch it happen, not just read it.
a² − b² = (a + b)(a − b)
The identity
A rectangle dressed up as squares.
The interactive figure
Use the arrows on the figure (or swipe) to follow the construction one piece at a time, then drag the sliders a and b (b always stays smaller than a), or switch from the «square» view to the «rectangle» view with the «View» switch. On a keyboard, ← and → change step.
Hover (or tap) a part of the drawing — the square, the corner, the rectangle — to find out what it is.
The mistake that gets mixed up with the square of a binomial
Some people write a² − b² = (a − b)², confusing it with the binomial identity. They are not the same thing: with a = 5 and b = 3, a² − b² is 16, but (a − b)² is only 4. The difference of squares is not a square. It is a product, and this figure shows why.
a² − b² = 40(a + b)(a − b) = 40a + b = 10a − b = 4
With a = 7 and b = 3: 7² − 3² = 40 = (7+3) × (7−3).
The construction, step by step
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Step 1 · The mistake
The mistake that gets mixed up with the square of a binomial
Some people write a² − b² = (a − b)², confusing it with the binomial identity. They are not the same thing: with a = 5 and b = 3, a² − b² is 16, but (a − b)² is only 4. The difference of squares is not a square. It is a product, and this figure shows why.
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Step 2 · The square
We start from a square of side a
Its area is a². From here we imagine removing a corner: a small square of side b, with b smaller than a, and what is left is a² − b², the quantity we care about.
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Step 3 · The corner removed
A b × b corner is removed
At the top right, a small square of side b and area b². What is left of the big square, the L-shaped form called a gnomon, has area a² − b² by construction: it is the square minus the corner.
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Step 4 · The gnomon
The gnomon is cut into two rectangular pieces
A single cut, at the same height as the removed corner, divides the L-shaped form into two rectangles: a narrow tall one (a − b) × b and a wide low one a × (a − b). Together they are still worth a² − b²: a cut does not change the total area.
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Step 5 · The rearrangement
The narrow piece slides next to the other
Rotated and moved, the (a − b) × b piece sits beside the other with no overlaps and no gaps: together they form a clean rectangle, a + b wide and a − b tall. Its area, (a + b)(a − b), is the same area as before: a² − b².
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Step 6 · Try it
Change a and b and watch the rectangle reassemble
Drag the sliders; b always stays smaller than a, otherwise there would be no gnomon to cut. For every combination, the area of the final rectangle and that of the starting gnomon always coincide.
What each piece of the drawing is
In the figure these explanations appear when you hover over the element. Here they are all written out.
- The square of side a
- The starting point, area a². The corner at the top right is about to be removed.
- The corner removed, b × b
- A small square of side b, area b². It is the quantity that is subtracted from a².
- The gnomon
- The L-shaped form that is left after removing the corner: the big square minus the small square. Its area is exactly a² − b², by construction, not by coincidence.
- The cut
- A single horizontal cut, at the same height as the removed corner, divides the gnomon into two rectangles without changing the total area.
- The rearranged rectangle
- a + b wide, a − b tall. Its area, (a + b)(a − b), is the same as that of the starting gnomon: that is why a² − b² = (a + b)(a − b).
The questions we always get.
Why is a² − b² not (a − b)²?
(a − b)² is (a − b) multiplied by itself: a² − 2ab + b². a² − b² is a completely different quantity: the −2ab is missing and the sign of b² is the opposite. They are two different identities that look alike only in the symbols they use.
Why does b have to be smaller than a in this figure?
Because the construction removes a b × b corner from an a × a square: if b were equal to or larger than a, the corner would cover the whole square (or more) and there would be no gnomon left to cut. The algebraic identity stays true even with b ≥ a, but it loses the meaning of «area that is left».
What is it used for in practice?
It is the quickest tool for factorising certain algebraic expressions and for rationalising denominators with radicals (multiplying by the «conjugate»). Recognising it at a glance, two squares with a minus sign between them, saves steps in many algebra and calculus exercises.
Keep going from here
Does algebraic factorising never add up?
The difference of squares keeps coming back: in factorisations, in rationalising radicals, in quadratic equations. Recognising it at a glance saves time for years.