Squaring numbers ending in 5: the Vedic method that takes 3 seconds
Most people write out four lines of long multiplication to square 85. The Vedic method does it in two steps: one multiplication and one append. The answer is 7225, and once you see why, you can't unsee it.
If someone asked you to square 85 right now — no paper, no calculator — how long would it take?
Most people set up long multiplication: 85 × 85, four lines of working, two minutes, one likely arithmetic error. The answer is 7225. A student who knows this Vedic shortcut for squaring numbers ending in 5 produces 7225 in 3 seconds by doing one multiplication and writing two digits. No long multiplication. No intermediate steps to lose track of.
This technique works for any number ending in 5 — two-digit, three-digit, four-digit. The rule is the same every time. Once you've seen it, you'll use it for the rest of your life.
The rule, stated simply
Take everything before the 5. Multiply it by the next number up. Write the result, then append 25. Done.
65² → take 6, next up is 7, so 6×7 = 42
Write 42, append 25 → 4225
Basic
Take the tens digit, multiply by the next number up, append 25.
65²
Tens digit: 6
Next up: 7
6 × 7 = 42
Append 25
Answer: 4,225
35²
Tens digit: 3
Next up: 4
3 × 4 = 12
Append 25
Answer: 1,225
Check: 35 × 35 = 1,225 ✓
Try it: 45². You should have the answer in under 5 seconds.
High 2-digit
Same rule — the tens digit just happens to be large. Carry if needed.
85²
Tens digit: 8
Next up: 9
8 × 9 = 72
Append 25
Answer: 7,225
95²
Tens digit: 9
Next up: 10
9 × 10 = 90
Append 25
Answer: 9,025
3-digit
The "everything before the 5" rule scales directly to 3-digit numbers.
105²
Before the 5: 10
Next up: 11
10 × 11 = 110
Append 25
Answer: 11,025
135²
Before the 5: 13
Next up: 14
13 × 14 = 182
Append 25
Answer: 18,225
Check: 135 × 135 = 18,225 ✓
Challenge
The rule works on any number ending in 5, no matter how large.
295²
Before the 5: 29
Next up: 30
29 × 30 = 870
Append 25
Answer: 87,025
125²
Before the 5: 12
Next up: 13
12 × 13 = 156
Append 25
Answer: 15,625
See the pattern in motion
The diagram below works through 65² step by step, showing exactly where each digit of the answer comes from.
[Interactive visual flow available at https://www.thevedicmind.com/blog/squaring-numbers-ending-in-5.]
Why squaring ending in 5 always works — the algebra behind the pattern
This isn't a coincidence or a trick that works for special cases. Any number ending in 5 can be written as (10n + 5), where n is the digits before the 5. Square it and you get:
(10n + 5)² = 100n² + 100n + 25
= 100n(n + 1) + 25
= [n × (n+1)] followed by 25
The Vedic sutra behind this is Ekadhikena Purvena — "by one more than the previous one." The name describes exactly what you do: take the previous digit (n) and multiply by one more than it (n+1). The 25 at the end is a fixed consequence of squaring any number ending in 5.
Standard long multiplication treats 65² as a series of steps: 5×5, then 5×6, then 6×5, then 6×6, then add. Six operations, two levels of working, two carry checks. The Vedic method does one multiplication and one append because it operates on the algebraic structure rather than the digits independently.
The next Vedic technique that uses the same logic is near-base multiplication — 97 × 98 in two steps. Same idea, applied to multiplication instead of squaring.
Stump someone today
Ask someone to square any number ending in 5 — 75, 145, 225, anything. Say the answer before they finish setting up the calculation. Then show them how.
The best version: pick 995². They expect long multiplication to be hard. You say 990,025 in five seconds. That's 99 × 100 = 9,900, append 25. The same two steps.
Try the squaring lesson free → — No signup required. Takes about 3 minutes.
https://www.thevedicmind.com/learn/lesson/l1-ekadhikena-squares