Estimating square roots
To estimate a square root, find the two square numbers on either side. √50 is between √49 = 7 and √64 = 8, so it is a bit more than 7.
The idea: use the squares you know
Most numbers are not perfect squares, so their square roots are not whole numbers. But you can always find which two whole numbers they sit between.
For √50, the nearest squares are 49 = 7² and 64 = 8². So √50 is between 7 and 8, and much closer to 7.
See it: pictures to draw with your child
Copy these onto paper or a whiteboard. A quick sketch makes the idea concrete before any written method.
Step 1: between which two whole numbers?
List the squares until you pass the number: 1, 4, 9, 16, 25, 36, 49, 64. For √30, it is between 25 and 36.
So √30 is between 5 and 6. This is enough for most grade 8 questions, and for placing roots on a number line.
Step 2: get one decimal place
Ask how far along the gap the number is. 50 is 1 of the 15 steps from 49 to 64, so √50 is about 7 + 1/15 ≈ 7.07.
Check by squaring: 7.1² = 50.41, which is just over 50. So √50 is a little under 7.1.
Check a guess by squaring it
Square your estimate. If 7.1² is just over 50, then √50 is just under 7.1. Try other guesses.
Roots that never end
√2 = 1.41421356… goes on forever and never repeats. Numbers like this are called irrational, and π is another one.
Grade 8 places them on a number line by estimating. √2 is between 1.4 and 1.5, because 1.4² = 1.96 and 1.5² = 2.25.
Ordering numbers with square roots
To put 7, √48 and √50 in order, compare their squares: 49, 48 and 50. So √48 < 7 < √50.
Squaring keeps the order of positive numbers, so this trick always works. It is much faster than working out decimals.
Key words
- Perfect square: a whole number with a whole-number square root, like 49.
- Estimate: a close answer, like √50 ≈ 7.1.
- Irrational number: a number whose decimals never end or repeat, like √2.
- Number line: a line with numbers in order, used to place roots.
Teaching tip: know the squares to 15²
Estimating is quick only if the squares are known. Spend two minutes a day on the squares from 1² to 15² until they are instant.
Then practice with a number line drawn from 0 to 15. Mark each estimate and check it by squaring on a calculator.
Children enjoy “closest wins”: both of you estimate √70, then check. Whoever is closer to 8.37 scores a point.
Over a few days, the estimates get sharper. Most children are soon within 0.1 of the real root, which is all grade 8 asks for.
A week of practice
Ten minutes a day is plenty. Short, regular practice beats one long session at the weekend.
- Day 1: the squares from 1² to 15² from memory.
- Day 2: between which two whole numbers, for ten different roots.
- Day 3: one decimal place, checked by squaring.
- Day 4: ordering numbers and roots on a number line.
- Day 5: the quiz on this page, then a round of estimating square roots practice.
If one day goes badly, repeat it rather than moving on. Each day builds on the one before.
Quiz: estimating square roots
Between which two whole numbers?
Where this fits: exponents step by step
What to say to your child
A short script for teaching it the first time. Pause after each line and let your child do the next step.
- “Is 50 a square number?” (No.)
- “Which squares are either side?” (49 and 64.)
- “So √50 is between…?” (7 and 8.)
- “Closer to which?” (7, because 50 is close to 49.)
- “How can we check?” (Square 7.1: 50.41.)
More worked examples
Example 1: between
√30 is between 5 and 6, because 25 < 30 < 36.
Example 2: one decimal
√20 ≈ 4.5, because 4.5² = 20.25.
Example 3: irrational
√2 is between 1.4 and 1.5.
Example 4: ordering
√48 < 7 < √50, because 48 < 49 < 50.
Practice questions (answers below)
Ask these out loud or write them down. Check each answer, and ask “how did you get that?” for one or two.
- √30 is between…
- √80 is between…
- √2 is between…
- √120 is between…
- √60 to one decimal place
- √17 to one decimal place
- √99 to one decimal place
- Which is bigger, √50 or 7?
- Order √48, 7, √50
- Stretch: √200 to one decimal place
Show the answers
- 5 and 6
- 8 and 9 (close to 9)
- 1 and 2
- 10 and 11
- 7.7
- 4.1
- 9.9 (9.95)
- √50
- √48, 7, √50
- 14.1
Common mistakes
How to teach it at home
Closest wins
Both estimate a root like √70, then check on a calculator. Closest scores a point.
Square spotting
Call out a number; your child names the two squares it sits between.
Tile test
Try to make 50 tiles into a square. It can’t be done, but 7 by 7 nearly works.
By grade
- Grade 8 (8.NS.A.2): use rational approximations of irrational numbers and place them on a number line.
- Grade 8 (8.EE.A.2): square roots of perfect squares, and roots as solutions of x² = p.
- In India, estimating square roots is in the class 8 chapter A Square and a Cube.
- Back to: square roots.
More for each grade: Grade 8 math and the CBSE class 8 maths guide.
Worksheets and practice
Square roots worksheet: printable PDF with an answer key. · Make a fresh test
Exponents practice · exponent calculator · exponents for kids · exponent rules worksheet · scientific notation worksheet · order of operations with exponents
Last reviewed
Questions parents ask
How do you estimate a square root?
Find the two square numbers either side. √50 is between √49 = 7 and √64 = 8, so it is about 7.1.
What is the square root of 50?
About 7.07. It is not a whole number, because 50 is not a perfect square.
How do you get a square root to one decimal place?
Estimate, then square your estimate to check. 7.1² = 50.41, so √50 is just under 7.1.
What is an irrational number?
A number whose decimals go on forever without repeating, like √2 or π.
What grade learns estimating square roots?
Grade 8 under Common Core 8.NS.A.2.