Squaring Tricks: Fast Methods, Formulas and Examples

Squaring Tricks use algebraic identities, base-10 methods and digit patterns to calculate squares quickly. This page covers the square of numbers ending in 5, numbers close to 10, 100 or 1,000, and numbers that can be split into convenient parts. Each method includes its formula, calculation rule and a simple example.

On this page

What Are Squaring Tricks?

Squaring tricks are shortcut methods for finding the product of a number by itself without using long multiplication. They are based mainly on identities such as (a + b)² = a² + 2ab + b² and (a − b)² = a² − 2ab + b².

A number can be expressed near a convenient base or divided into two simple parts. For example, 47² can be calculated as (40 + 7)² = 40² + 2 × 40 × 7 + 7² = 2,209. The selected method depends on the number's last digit, its distance from a base, or the ease of splitting it.

Squaring Tricks Formula & Tricks

Important Formulas

Square of a sum
(a + b)² = a² + 2ab + b²

Use this when a number can be written as the sum of two convenient numbers.

Square of a difference
(a − b)² = a² − 2ab + b²

Use this when a number is slightly less than a convenient round number.

Number ending in 5
(10a + 5)² = a(a + 1) × 100 + 25

Multiply the part before 5 by the next integer, append 25, and obtain the square.

Number near a base
(B ± d)² = B² ± 2Bd + d²

B is a convenient base such as 10, 100 or 1,000, and d is the difference from that base.

Quick Tricks

Square a number ending in 5

For a number of the form 10a + 5, multiply a by a + 1 and append 25.

Example: 85²: 8 × 9 = 72, so 85² = 72|25 = 7,225.
Use the nearest power-of-10 base

Find the difference from 10, 100 or 1,000. Add or subtract twice the base-distance term, then add the distance squared.

Example: 98² = (100 − 2)² = 10,000 − 400 + 4 = 9,604.
Use the base concatenation method

For numbers close to 100, calculate the left part as 100 ± 2d and append d² as a two-digit block.

Example: 103²: d = 3, so the left part is 100 + 6 = 106 and the right part is 09. Therefore, 103² = 10,609.
Split the number into tens and units

Write the number as a + b and apply (a + b)². Choose parts that make multiplication simple.

Example: 63² = (60 + 3)² = 3,600 + 360 + 9 = 3,969.

Squaring Tricks Concepts

Squaring Numbers Ending in 5

A number ending in 5 can be squared by multiplying the digits before 5 by the next integer and appending 25.

If the number is 10a + 5, then its square is a(a + 1) × 100 + 25. The product a(a + 1) forms the left part, while 25 is always the last two digits.

Example: 115²: 11 × 12 = 132, so 115² = 13,225.

Squaring Numbers Just Above a Base

For a number B + d, use (B + d)² = B² + 2Bd + d².

Choose B as 10, 100 or 1,000. For a number close to 100, the calculation can be written as (100 + d)² = 10,000 + 200d + d². For 100 + d, the concatenation form is (100 + 2d)|d² when d² is written using two digits.

Example: 107² = (100 + 7)² = 10,000 + 1,400 + 49 = 11,449.

Squaring Numbers Just Below a Base

For a number B − d, use (B − d)² = B² − 2Bd + d².

The middle term is subtracted, but the final d² term is always added. For numbers below 100, write (100 − d)² = 10,000 − 200d + d². In the concatenation method, the left part is 100 − 2d and d² is padded to two digits.

Example: 94² = (100 − 6)² = 10,000 − 1,200 + 36 = 8,836.

Squaring by Splitting into Convenient Parts

Any number can be written as a + b and squared using a² + 2ab + b².

This method is useful when the tens or hundreds part has an easy square. The cross-product term 2ab must be included; omitting it gives an incorrect result.

Example: 124² = (120 + 4)² = 14,400 + 960 + 16 = 15,376.

Squaring Tricks Video Lessons

Watch short topic-wise lessons for quick revision.

8 Lessons
Lesson 1 of 8 Quick Revision

Vedic Maths Squaring Tricks

Learn Vedic Maths techniques for calculating squares efficiently, with clear steps for applying squaring tricks to suitable numbers in quantitative aptitude problems.

Continue with more lessons and practice in PrepShots.Watch More in App - Start ₹1 Trial →
More Squaring Tricks Lessons Scroll to explore →

Practice Squaring Tricks Questions

Practise published questions related to this topic.

1Find the maximum number of students among whom 165 chips and 385 toffees can be distributed such that each student gets the same number of each.→ 2A dealer purchased a laptop for ₹99,000. He allows a discount of 37% on its marked price and still gains 40%. Find the marked price of the laptop.→ 3Rajesh has bought a watch for ₹5,000 from a local store and plans to sell it to a friend at a 20% discount on the marked price. Rajesh wants to make a 25% profit on the cost price after the discount. How much will he sell it for?→ 416 m, 20 m and 24 m are the dimensions of a room. Find the length (in m) of the greatest possible scale to measure all dimensions exactly.→ 5The LCM of two numbers is 84. The numbers are in the ratio 4 : 3. Find the sum of the numbers.3129→ 6If √(x+1) = 5 , then what is the value of x?→ 7The largest four-digit number which when divided by 12, 13 and 7 leaves remainder 3 in each case is:→ 8A person sells an article at the loss of 25 percent. Cost price and selling price are increased by 40 percent and 'y' percent respectively. If the new loss percentage is 15 percent, then what is the value of 'y'?→ 9A trader buys a consignment of goods for ₹2400. He sells one-third​ of the goods at a 20% profit, and one-fourth​ of the goods at a 10% loss, and the remaining goods at cost price. What is the overall profit or loss percentage?→ 10During a sale, 55% of the goods are sold at a 41% profit, 20% of the remaining goods are sold at a 19% profit and the rest are sold at a 21% loss. If there is an overall profit of x%, then what is the value of x?→

Squaring Tricks Quick Quiz

Attempt 5 questions and check your score instantly.

Quick Revision Notes

Squaring Tricks Revision Points

Use the method that matches the number's form and check the middle term and digit placement carefully.

  • (a + b)² = a² + 2ab + b².
  • (a − b)² = a² − 2ab + b².
  • For a number ending in 5, use a(a + 1) × 100 + 25.
  • For B ± d, calculate B² ± 2Bd + d².
  • In base-100 concatenation, pad d² to two digits; for base 1,000, pad it to three digits.
  • The final d² term is added in both the plus and minus forms.
  • When splitting a number, never omit the cross-product term 2ab.

Squaring Tricks FAQs

What is the shortcut formula for squaring a number ending in 5?

For (10a + 5)², use a(a + 1) × 100 + 25. For example, 65² = 6 × 7 followed by 25 = 4,225.

How can 99² be calculated using a base-100 method?

99 = 100 − 1, so 99² = 10,000 − 200 + 1 = 9,801. In concatenation form, 100 − 2 = 98 and 1² is written as 01, giving 98|01.

Why is zero added in the base-100 method for 97²?

The difference is 3 and d² = 9. Since the right block must contain two digits, 9 is written as 09. Thus, 97² = (100 − 6)|09 = 94|09 = 9,409.

How do you square 125 using splitting?

Write 125 as 100 + 25: 125² = 10,000 + 2 × 100 × 25 + 625 = 15,625.

What is the difference between (a + b)² and a² + b²?

(a + b)² includes the cross-product term: a² + 2ab + b². Therefore, (3 + 4)² = 49, while 3² + 4² = 25.

Continue learning Squaring Tricks on PrepShots

Continue on PrepShots