Three sides

The idea

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca, obtained by replacing b + c by d in (a + d)² and expanding.

Three pieces on a side

Now each side of a square is made of three pieces: a, then b, then c. So the side is a + b + c. Draw lines across and down at all the joins.

Into how many pieces does the square fall?

• 6
• 9
• 12

Cut it up

Change a, b and c. The pieces are squares and rectangles. Check that they add up to the whole square. Try three different settings.

Three squares, six rectangles

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca: three squares and six rectangles, two of each kind.

Replace b + c by d: (a + d)² = a² + 2ad + d². Put d = b + c back: a² + 2a(b + c) + (b + c)² = a² + 2ab + 2ac + b² + 2bc + c².

Notes

(a + b + c)² = a² + b² + c² + 2ab + 2bc + 2ca. A square of side a + b + c has three squares and six rectangles.

Check yourself

A square of side a + b + c is cut into pieces. How many of the pieces are rectangles (not squares)?

Answer: 6

Three squares and 6 rectangles: two of ab, two of bc and two of ca.

For a = 1, b = 2, c = 3, what is 2ab + 2bc + 2ca?

Answer: 22

2 × 2 + 2 × 6 + 2 × 3 = 4 + 12 + 6 = 22. With a² + b² + c² = 14, the whole is 36 = 6².

Find 119² using (100 + 10 + 9)².

Answer: 14161

10000 + 100 + 81 + 2000 + 1800 + 180 = 14161.

Which is the expansion of (x + y + 1)²?

• x² + y² + 1. The six rectangles are missing.
• x² + y² + 1 + xy + y + x. Each rectangle appears twice, so the cross terms are 2xy, 2y and 2x.
• x² + y² + 1 + 2xy + 2y + 2x — correct. Yes: three squares, then 2 × xy, 2 × y × 1 and 2 × x × 1.