What is the square of a trinomial identity?

The square of a trinomial identity states that (a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc. Squaring a sum of three terms produces six terms: three squared terms and three doubled cross products, one for each pair of original terms.

Why are there exactly six terms in the expansion?

A 3x3 grid has nine cells, but the three off-diagonal cross-product pairs are equal: ab equals ba, ac equals ca, and bc equals cb. So the nine cells collapse to six distinct values: three diagonal squared terms and three doubled cross products.

How does this generalize the (a+b)^2 identity?

The square-of-a-sum builds a 2x2 grid: two squares (a^2, b^2) and two equal ab rectangles producing 2ab. The trinomial extends this to a 3x3 grid: three squares on the diagonal and three doubled cross products. The pattern continues: squaring a sum of n terms produces an n-by-n grid with n diagonal squares and n(n-1)/2 doubled cross products.

How do I expand expressions like (x+2y+3)^2?

Apply the pattern: square each term, then add twice the product of every pair. So (x+2y+3)^2 = x^2 + (2y)^2 + 3^2 + 2(x)(2y) + 2(x)(3) + 2(2y)(3) = x^2 + 4y^2 + 9 + 4xy + 6x + 12y.

What does the explosion view at the end show?

After step 3, the nine grid pieces oscillate outward from the center and back. Pulled apart, you can see each of the nine cells as a distinct piece and verify that the pairs ab, ac, and bc each appear twice. The motion makes the symmetry of the 3x3 layout visible without the cells being crowded together.

Square of a Trinomial (a+b+c)²

Speed

The (a+b+c)² Identity at a Glance

Reading the Starting Square

Splitting Each Side into Three Segments

Building the 3×3 Grid

The Explosion View

Why Each Cross Product is Doubled

Using the Controls

Related Concepts and Tools

Visualizing (a+b+c)² as a 3×3 Grid

This visualizer extends the square-of-a-sum dissection to three terms. A square of side

a+b+c

is split into nine rectangles arranged as a 3×3 grid: three squared terms on the diagonal and six cross-product rectangles in three matched pairs. The full identity

(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

falls out of summing the nine pieces — and the explosion view at the end separates them so each one can be seen on its own.

The (a+b+c)² Identity at a Glance

The square of a trinomial identity expands the square of a three-term sum:

(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

Three squared terms cover the diagonal of the expansion. Three cross-product terms cover the off-diagonal pairs, each appearing with a coefficient of

2

. Six terms total — fewer than the nine you might initially expect, because cross products like

ab

and

ba

are equal and combine.

This identity is the natural step beyond

(a+b)^2

. The same dissection logic that proves the binomial case extends with no modification: more terms, more pieces, same area-conservation argument.

The proof unfolds in four frozen frames below: reading the starting square, splitting each side into three segments, building the 3×3 grid, and the explosion view.

Reading the Starting Square

Step 1 of the animation shows a square of side

a+b+c

, with

(a+b+c)^2

floating as a centred label and dimension marks "

a+b+c

" along the top and left edges. The square's area is

(a+b+c) \times (a+b+c) = (a+b+c)^2

— the quantity to be decomposed.

The starting frame is intentionally similar to the

(a+b)^2

frame, just with one more term in the side length. The next steps will show that the structural argument is identical to the binomial case, only with a 3×3 grid replacing the 2×2 grid.

Step 1: the (a+b+c) square, frozen

One more term in the side (a = 4, b = 2, c = 3), same opening move: a single area with a compound name.

The visualizer picks

a = 4

b = 2

c = 3

— three visibly different segment lengths, so the nine future cells will all have distinct, readable proportions. Deliberately unequal segments matter here more than in the binomial case: with nine cells coming, equal splits would make the grid look like a tic-tac-toe board and hide which cell is which product.

One extra term costs nothing structurally, and that is this frame's real message: the side is longer, the square is bigger, and the argument ahead — split, cut, add — is unchanged.

Splitting Each Side into Three Segments

Step 2 splits each side of the square into three labelled segments:

a

, then

b

, then

c

. Both the top and left edges show this three-way split, and the dimension labels rearrange to display each segment separately while still showing the full "

a+b+c

" total.

The labels on the top and left edges read

a

b

c

in the same order. This consistent labelling is essential for the next step: when the grid lines are drawn, the cell at row

i

, column

j

will have dimensions equal to the

i

th segment by the

j

th segment — and the labelling makes those dimensions immediately readable.

No area has been added or removed. The square is still the same square; only its boundary description has been refined.

Step 2: the three-way split, frozen

Each edge divided a, b, c with the totals slid outward — the bookkeeping that will name all nine cells.

The frozen frame shows six small dimension labels —

a

b

c

twice over — plus the two combined "

a+b+c

" labels slid outward, all coexisting. That is the same double-bookkeeping the binomial proof used, scaled up: the outer labels hold the left-hand side of the identity, the inner ones are about to generate the right.

Row-and-column indexing starts paying off in the next frame: cell

(i, j)

will have the

i

th segment for height and the

j

th for width, no further thought required.

Building the 3×3 Grid

Step 3 fills in the dissection. Two vertical lines drop from the

a

b

and

b

c

tick marks on the top edge. Two horizontal lines extend from the

a

b

and

b

c

tick marks on the left edge. Together they partition the square into nine rectangles arranged as a 3×3 grid.

Each cell takes a colour and a label according to its dimensions:

• Diagonal cells (top-left, centre, bottom-right) are squares with sides

a

b

c

respectively. They are labelled

a^2

b^2

c^2

, each in a unique colour.

• Off-diagonal cells are rectangles. The top-middle and middle-left both have dimensions

a

and

b

and are labelled

ab

. The top-right and bottom-left both have dimensions

a

and

c

and are labelled

ac

. The middle-right and bottom-middle both have dimensions

b

and

c

and are labelled

bc

. Each off-diagonal pair shares a colour to emphasise that the two cells in a pair are equal.

Step 3: the 3×3 grid, frozen

Nine cells in six colours: three squares on the diagonal, three mirrored pairs off it. Commutativity as symmetry.

The frozen grid rewards a minute of deliberate reading: find the diagonal (

a^2

b^2

c^2

in blue, pink, green), then verify each off-diagonal colour appears exactly twice, mirrored across the diagonal. That mirror symmetry is the commutativity of multiplication drawn as geometry — cell

(i,j)

and cell

(j,i)

are the same rectangle rotated.

The six colours for six distinct products also preview the answer's shape: six terms, three of them doubled. The explosion view pulls the nine pieces apart so the count can be made piece by piece.

The Explosion View

Step 4 visually separates the nine pieces with an oscillating explosion. Every cell drifts outward from the centre of the square, gaps opening between rows and columns, then drifts back together. The motion repeats so the layout can be read in either configuration.

Pulled apart, the nine cells become individually inspectable. The three diagonal squares stand out:

a^2

b^2

c^2

in their distinct colours. The three off-diagonal pairs now read clearly as twin pieces: two

ab

rectangles, two

ac

rectangles, two

bc

rectangles, each pair sharing a colour. Adding all nine areas:

a^2 + b^2 + c^2 + ab + ab + ac + ac + bc + bc

= a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

The total area equals

(a+b+c)^2

from the original square, so:

(a+b+c)^2 = a^2 + b^2 + c^2 + 2ab + 2ac + 2bc

Step 4: the explosion, frozen at full separation

The nine pieces pushed apart along their own rays — count them: a² + b² + c² and three doubled cross products.

The frozen frame holds the explosion at full separation — each cell pushed outward along its own line from the centre, gaps open, every label readable. This is the counting configuration: nine separate pieces, and the tally

3 + 3 \times 2

terms performable by pointing.

A generalization is visible from here at no extra cost: an

n

-term sum squared would give an

n \times n

grid —

n

diagonal squares and

\binom{n}{2}

mirrored pairs. The trinomial is simply the first case where the grid is big enough to make that pattern unmistakable.

Why Each Cross Product is Doubled

The 3×3 grid has nine cells but produces only six distinct expression values. The reason is symmetry: the cell at row

i

, column

j

has dimensions equal to the

i

th and

j

th segments, but the cell at row

j

, column

i

has the same dimensions in the opposite orientation. They are different geometric pieces in different positions, but they have the same area.

Specifically:

• Top-middle cell has dimensions

a \times b

. Middle-left cell has dimensions

b \times a

. Same area

ab

.

• Top-right has dimensions

a \times c

. Bottom-left has dimensions

c \times a

. Same area

ac

.

• Middle-right has dimensions

b \times c

. Bottom-middle has dimensions

c \times b

. Same area

bc

.

Three pairs of equal cells produce the three doubled cross products. The symmetry is purely geometric — it follows from the commutativity of multiplication,

xy = yx

, manifested as area-preserving rotation of a rectangle.

Using the Controls

Five controls drive the animation:

• Play runs the full proof. After step 3 fills the grid, step 4 starts the perpetual explode-and-reform oscillation that reveals the nine pieces individually.

• Pause stops the current animation, including the stage-4 oscillation, in place.

• Step Forward and Step Back move through the four steps. Forward from step 3 starts the oscillation; Back from step 3 returns to the unfilled grid.

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

• Speed controls all transitions and the explosion oscillation, from 0.5× to 2×. The grid is more readable at slower speeds; the oscillation is more dramatic at faster speeds.

The right panel lists the four written steps with the current step highlighted.