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Algebraic Identities



Square of a Sum
(a + b)²
= a² + 2ab + b²
Watch a square split into a 2×2 grid of four rectangles. The two identical ab pieces sit at opposite corners — the source of the doubled middle term. The cleanest entry point if you have not seen this style of proof before.
Open visualizer →Read more ↓
Square of a Difference
(a − b)²
= a² − 2ab + b²
Two ab strips cover an a-square but overlap on a small b² corner. The discard step makes the negative middle term unmistakable — the minus sign is geometry, not a rule to memorise.
Open visualizer →Read more ↓
Square of a Trinomial
(a + b + c)²
= a² + b² + c² + 2ab + 2ac + 2bc
The binomial-square argument extended to three terms. A 3×3 grid produces nine pieces — three squared terms on the diagonal and three pairs of cross-products that explode outward in the final view.
Open visualizer →Read more ↓
Difference of Squares
a² − b²
= (a + b)(a − b)
A small square is cut from a larger one. The L-shape that remains slices into two rectangles, and one rotates 90° to fit beside the other — forming a rectangle whose sides are the two factors. Geometric factoring at its clearest.
Open visualizer →Read more ↓


What These Visualizers Do
Learning by Watching Areas Move
Step Through at Your Own Pace
Pick a Visualizer
Related Concepts and Tools




What These Visualizers Do

Each visualizer in this collection turns one algebraic identity into an animated geometric proof. A square is built, marked, cut, and rearranged step by step. At the end, the same area is described two ways — and that double description is the identity.

Every tool follows the same four-step rhythm: a starting square that represents the identity's left-hand side, a labelling step that splits sides into named segments, a dissection or rearrangement that physically reorganises the area, and a summing step that adds up the pieces to produce the right-hand side. Across the four identities the specifics differ — what gets cut, what overlaps, what rotates — but the underlying argument is always the same: total area is preserved, only its arrangement changes.

The animation is paired with a written step panel that updates as you move forward. By the time you reach the final step, the equation has been built piece by piece, and you can read it directly off the diagram instead of recalling it from memory.

Learning by Watching Areas Move

Algebraic identities are usually presented as formulas to memorise. The visual approach replaces memorisation with a physical argument that you can watch happen. Two ideas do most of the work:

• Area is conserved under cutting and rearranging. Slice a square into pieces and slide them around — the total area never changes. So if a shape can be described two different ways before and after, those two descriptions must be equal as numbers.

• Each term in the expansion corresponds to a specific piece. Instead of 2ab2ab being "a coefficient with a 2 in front", it becomes two visibly identical rectangles sitting at opposite corners. Instead of −2ab-2ab being a sign rule, it becomes two strips that overlap on a patch counted twice. The sign and the coefficient stop being arbitrary — they are forced by what you see.

This is why the visual proofs tend to stick. You are not remembering a formula; you are remembering a picture, and the formula is what the picture says.

Step Through at Your Own Pace

Every visualizer ships with the same set of controls so you can watch the proof in whatever way suits you best:

• Play runs the entire animation end to end. Useful for a first overview before going deeper.

• Step Forward / Step Back advance or reverse one transition at a time. Useful when you want to study a specific move — a rotation, a colour fill, an overlap — without it slipping past.

• Pause halts whatever is moving without losing position, so you can stop on a frame that matters.

• Speed ranges from 0.5× to 2×. Slower for first viewings; faster for review.

• Reset returns to step 1 and replays the intro fade-in.

This is why the visualizers reward repeated viewing. The first pass at default speed gives you the overall shape of the proof. A second pass stepping through manually lets you stop on each transition and ask "where did that piece come from? where did it go?" By the third pass the proof has stopped being a sequence of moves and started being a single object you understand all at once.

Pick a Visualizer

Each visualizer has its own dedicated page with a deeper write-up of how the proof works, what the controls do, and why the geometry matches the algebra. Use the brief descriptions below to pick a starting point.

Square of a Sum — (a+b)2=a2+2ab+b2(a+b)^2 = a^2 + 2ab + b^2. The cleanest entry point: a square is sliced by a 2×2 grid into four rectangles, and the four areas are read off. Best place to start if you have not seen this style of proof before. Open Square of a Sum visualizer.

Square of a Difference — (a−b)2=a2−2ab+b2(a-b)^2 = a^2 - 2ab + b^2. Two strips cover the square but overlap on a small corner; the discard step explains why the middle term is negative and the last term is positive. Best for understanding sign behaviour. Open Square of a Difference visualizer.

Square of a Trinomial — (a+b+c)2=a2+b2+c2+2ab+2ac+2bc(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc. Extends the binomial-square argument to three terms with a 3×3 grid. The final explosion view separates the nine pieces. Best after you are comfortable with the binomial case. Open Square of a Trinomial visualizer.

Difference of Squares — a2−b2=(a+b)(a−b)a^2 - b^2 = (a+b)(a-b). A small square is removed from a larger one; the resulting L-shape is sliced and rotated into a rectangle whose sides are exactly the two factors. Best for understanding factoring geometrically. Open Difference of Squares visualizer.

Related Concepts and Tools

These visualization tools connect to broader algebra concepts and resources:

Theoretical Foundations:

• Polynomials — Polynomial structure, terms, degrees, and standard operations.

• Factoring — Common polynomial factorings including the difference of squares and perfect square trinomials.

• Exponents and Powers — Power rules and the laws of exponents that underlie squaring and higher powers.

Reference Tools:

• Powers Table — Searchable reference of integer powers, useful for spotting perfect squares and computing (a+b)2(a+b)^2 for specific values.

More Visual Tools:

• Algebra Visual Tools — Index of all interactive algebra visualizations and proofs, including this identity collection.