Estimating Square Roots –

Estimating square roots is a non-calculator skill that appears on both Foundation and Higher GCSE Maths papers. You need to find which two consecutive whole numbers a square root lies between, and sometimes give a closer estimate.

What Is Estimating a Square Root?

When a number is not a perfect square, its square root is an irrational number — a decimal that goes on forever without repeating. On a non-calculator paper, you cannot find the exact value, but you can estimate it by identifying the two perfect squares it falls between.

For example, √50 lies between √49 = 7 and √64 = 8, so √50 is between 7 and 8. Since 50 is much closer to 49 than to 64, a good estimate is about 7.1.

This skill tests your knowledge of square numbers and your ability to reason about where a value sits between two known points.

Key Formulas

Step-by-Step Method

1. Identify the two consecutive perfect squares that the number lies between.
2. State the two whole number square roots — the answer lies between these.
3. For a closer estimate, see how far the number is between the two perfect squares and adjust proportionally.

Worked Example 1 — Foundation Level

Question: Estimate √40 to one decimal place.

Working:

Step 1 — 6² = 36 and 7² = 49. Since 36 < 40 < 49, √40 is between 6 and 7.

Step 2 — 40 is 4 above 36. The gap between 36 and 49 is 13. 4/13 ≈ 0.3.

Step 3 — Estimate: 6 + 0.3 = 6.3.

Answer: √40 ≈ 6.3

Worked Example 2 — Higher Level

Question: Without a calculator, estimate √110 and state which whole number it is closest to.

Working:

Step 1 — 10² = 100 and 11² = 121. Since 100 < 110 < 121, √110 is between 10 and 11.

Step 2 — 110 is 10 above 100. The gap between 100 and 121 is 21. 10/21 ≈ 0.48.

Step 3 — Estimate: 10 + 0.48 ≈ 10.5.

Answer: √110 ≈ 10.5, closest to 10 or 11 (almost exactly halfway, but slightly closer to 10).

Worked Example 3 — Exam Style

Question: Show that √75 lies between 8.6 and 8.7.

Working:

Step 1 — Calculate 8.6² = 73.96.

Step 2 — Calculate 8.7² = 75.69.

Step 3 — Since 73.96 < 75 < 75.69, we have 8.6 < √75 < 8.7.

Answer: √75 lies between 8.6 and 8.7 (shown).

Common Mistakes

• Using the wrong pair of perfect squares. Make sure you identify the correct consecutive squares. Memorise all squares up to 15² = 225.
• Forgetting that estimation is approximate. The linear interpolation method gives a reasonable estimate but not the exact value.
• Mixing up square roots and halving. √36 = 6, not 18. The square root asks "what number times itself gives 36?"

Exam Tips

• Learn your square numbers up to at least 15² = 225 so you can identify the surrounding squares quickly.
• For "show that" questions, you must square both boundary values and demonstrate the number lies between them.
• A common question format asks you to place √n on a number line — use estimation to position it accurately.

Practice Questions

Q1 (Foundation): Between which two consecutive whole numbers does √60 lie?

Q2 (Foundation): Estimate √20 to one decimal place.

Q3 (Higher): Show that √90 is between 9.4 and 9.5.

Practise estimating square roots questions with instant feedback — completely free on GCSEMathsAI.

Related Topics

• Square Numbers and Cube Numbers
• Powers and Roots
• Surds

Summary

• To estimate a square root, find the two consecutive perfect squares either side of the number.
• The square root lies between the roots of those two perfect squares.
• Use linear interpolation for a more precise estimate.
• For "show that" questions, square both boundary values and show the number falls between them.