Squaring a number means multiplying it by itself, so 7² is 7 × 7, which is 49. Squares turn up in area, in algebra, in Pythagoras' theorem and in square roots, so being able to square two-digit numbers quickly pays off again and again. This guide shows the patterns that make it fast, explains why each one works, and ends with a practice set.
If you don't yet know the squares from 1 to 20 by heart, start with our squares 1 to 20 guide. The tricks below build on them.
The square of a number ending in 5 always ends in 25. For the digits in front, multiply the tens digit by the next number up.
Why it works: 65 × 65 is the same as 60 × 70 plus 5 × 5. And 60 × 70 = 4,200, which is where the 6 × 7 comes from, then 5 × 5 adds the 25.
For a number just below 100, find how far below it is. Take that gap away from the number to get the hundreds, then add the square of the gap.
Consider squaring 98, which is 2 less than 100. Take the 2 away from 98 to get 96, which gives the first part: 96 × 100 = 9,600. Then add the square of the difference: 2 × 2 = 4. So 98² = 9,600 + 4 = 9,604. Check: 98 × 98 = 9,604.
For 97, the gap is 3: 97 − 3 = 94 and 3 × 3 = 9, so 97² is 9,409. Write the square of the gap as two digits, 09, so the answer has the right shape.
Above 100, add the gap instead. For 103, the gap is 3 above: 103 + 3 = 106, so 106 × 100 = 10,600, and 10,600 + 9 = 10,609.
The Vedic Maths version of this method, which also works near 1,000, is in Vedic Maths square tricks.
A similar trick works around 50, because 50 squared is 2,500. Add the gap to 25 (or take it away), use that as the hundreds, then add the square of the gap.
For any two-digit number, use the pattern (a + b)² = a² + 2ab + b². Take 36²: this can be represented as (30 + 6)² = 30² + 2 × 30 × 6 + 6². Calculate each part separately: 30² = 900, 2 × 30 × 6 = 360, and 6² = 36. Add them together: 900 + 360 + 36 = 1,296, giving you the square of 36.
When the ones digit is big, it is often easier to go down from the next ten with (a − b)² = a² − 2ab + b². For 49², think of 50 − 1: that gives 2,500 − 100 + 1, which is 2,401.
These patterns are algebra at work. If they look unfamiliar, our algebra basics guide explains how brackets multiply out.
Each square is the one before it plus the two numbers involved. So once you know 30² is 900, then 31² is 900 + 30 + 31, which is 961.
A related shortcut works for two numbers the same distance either side of a round number: their product is the round number squared, minus the distance squared. So 41 × 39 is 40² − 1, which is 1,599. Our guide to mental multiplication has more shortcuts like this.
Answers: (1) 1,225, from 3 × 4 = 12. (2) 7,225, from 8 × 9 = 72. (3) 9,216: the gap is 4, 96 − 4 = 92, and 4 × 4 = 16. (4) 10,816: 104 + 4 = 108, and 4 × 4 = 16. (5) 2,704: 25 + 2 = 27, and 2 × 2 = 4. (6) 5,041: 4,900 + 140 + 1. (7) 841: 900 − 60 + 1. (8) 2,499, which is 50² − 1.
Once squaring is quick, square roots become quick too; see fast square root calculations. To keep the multiplication facts behind every square sharp, play the multiplication game or use the times tables chart.
If it ends in 5, multiply the tens digit by the next number and write 25 after it. If it is near 50 or 100, use the near-base trick. Otherwise, split it into tens and ones: square each part and add twice their product.
A number like 65 is halfway between 60 and 70, and 65 × 65 = 60 × 70 + 25. Multiplying the tens digits 6 and 7 gives the 4,200, and the 25 is always added on the end.
Yes, especially near 100 or 1,000. For 103, add the gap of 3 to get 106 hundreds, then add 3 squared: the answer is 10,609.