Estimating Square Roots – Finding Roots That Aren't Perfect

Most whole numbers are not perfect squares – √40 doesn't come out to a tidy whole number the way √36 does. But you can still estimate it accurately without a calculator, just by using the perfect squares from the last few pages as reference points: find the two perfect squares your number sits between, and you instantly know the two whole numbers its square root sits between too.

The Estimation Method

Find the nearest perfect square below and above your number. The square root lies between their square roots.

Between which two whole numbers does √54 lie?

49 (7²) < 54 < 64 (8²), so √54 is between 7 and 8.

Rounding to the Nearest Whole Number

Round √54 to the nearest whole number.

√54 ≈ 7.35, which is closer to 7. Rounded, √54 ≈ 7.

Real-Life Application

Quick mental estimates: sanity-checking a calculator result.
Construction: estimating diagonal measurements on site without a calculator.
Exams: estimating square roots is a common non-calculator skill test.

Key Takeaways

To estimate √n, find the perfect squares directly below and above n.
The square root of n lies between the square roots of those two perfect squares.
Radicals, covered next, formalise the general notation used throughout roots and radicals.

Practice: Estimating Square Roots

Estimate the Square Root