How to Square Numbers Ending in 5
Level 2 Mental multiplication · Reviewed by The NumThink editorial team · Last reviewed
Short version: For a number ending in 5: multiply the leading part by itself-plus-one, then append 25. 85² = 8 × 9 hundreds + 25 = 7,225.
Every number ending in 5 shares a fingerprint: its square always ends in 25, and the digits in front are a simple a×(a+1). Algebra guarantees it — you just collect the reward.
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Write the number as 10a + 5. Then (10a + 5)² = 100a² + 100a + 25 = 100 × a(a + 1) + 25.
So the hundreds part is a(a+1) and the last two digits are always 25. For 85: a = 8, so 8 × 9 = 72 hundreds → 7,225.
The steps
- Drop the final 5. 85 becomes 8.
- Multiply by one more. 8 × 9 = 72.
- Append 25. 72 followed by 25 → 7,225.
Worked examples
easy 35²
Show the thinking
a = 3
3 × 4 = 12
Append 25: 1,225.
Answer: 1,225
medium 85²
Show the thinking
a = 8
8 × 9 = 72
Append 25: 7,225.
Answer: 7,225
hard 115²
Show the thinking
a = 11
11 × 12 = 132
Append 25: 13,225.
Answer: 13,225
When to use it
- The square of any number ending in 5 — 15 up to 995 and beyond.
- Nearby squares: 85² = 7,225 is a great anchor, because 84² = 7,225 − 85 − 84 = 7,056.
When not to use it
- Squares NOT ending in 5 — the rule has nothing to attach to.
- Products of two different numbers ending in 5 (like 35 × 45) — related but different algebra; do not apply this rule there.
Common mistakes
Using a×a instead of a×(a+1). 85² answered as 6,425 (from 8 × 8). Fix: The rule is a times a-plus-one — 8 × 9, not 8 × 8.
Losing the 25. Appending only 5, or forgetting the two-digit 25 entirely. Fix: The final two digits are always exactly 25 — that is the fingerprint of this rule.
Check yourself: Every square of a number ending in 5 ends in 25. Also estimate: 85² must sit between 80² = 6,400 and 90² = 8,100.