Squares by a round-number shortcut
Find 47² in your head. Make one factor a round number, one that ends in 0.
Step 1 of 5
- A square of side a = 8. Its area is 64.
- Cut it into 8 × 5 above and 8 × 3 below.
- Split the strip: a 3 × 3 square and a 5 × 3 piece.
- Turn the 5 × 3 piece. Put it beside the 8 × 5 piece.
- Now there is one 11 × 5 rectangle and a 3 × 3 square. So 64 = 55 + 9.
The rule
Rule: a² = (a + b)(a − b) + b². Choose b to make a factor round.
Here a is the number to square. And b is a number we choose. The rule comes from a² − b² = (a + b)(a − b). Add b² to both sides. It holds for every b. A good b makes one factor a multiple of 10. Then the product is easy.
For 47², take a = 47 and b = 3. Then a + b = 50 and a − b = 44. So 47² = 50 × 44 + 3² = 2200 + 9 = 2209.
| Number | b | a + b, a − b | Product + b² | Square |
|---|---|---|---|---|
| 47 | 3 | 50, 44 | 2200 + 9 | 2209 |
| 82 | 2 | 84, 80 | 6720 + 4 | 6724 |
| 93 | 7 | 100, 86 | 8600 + 49 | 8649 |
| 118 | 2 | 120, 116 | 13920 + 4 | 13924 |
Change b. Find a b that makes a factor round.
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