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B1.2 · Identify perfect squares and find their square roots

Learn to identify perfect squares and find their square roots through clear examples and targeted practice.

Ontario Grade 7 Mathematics

Number

Build a square before taking its root

A perfect square is a whole number that can be arranged in equal rows and columns to make a square. The picture helps us see why 36 is a perfect square: six rows of six unit squares make 36. A square root reverses this question. If the area of a square is known, its square root gives the positive side length. This is a model made from equal unit squares, not a photograph of a measured object.

What you will learn

  • Recognize perfect squares with a square array.
  • Find a positive square root of a perfect square.
  • Check a side length by multiplying it by itself.

Build equal rows and columns

A 6 by 6 array has six rows, and each row contains six equal cells. Count 6 × 6 = 36 cells. A rectangular 4 by 9 arrangement also has 36 cells, but it does not explain the side of a square.

Reverse the area calculation

The area of a square is side times side. To find the side from area 36 square centimetres, look for a positive number that multiplied by itself equals 36. That number is 6.

Keep units and signs clear

Area has square units, while a length has ordinary units. The symbol √36 names the principal, nonnegative square root 6. The equation x² = 36 also has a negative solution, but a physical side length cannot be negative.

Worked example

Find the side of a square tile

A square tile has area 36 cm². Find its side length.
Six equal rows and six equal columns
Six equal rows and six equal columnsThe 36 equal cells form a 6 by 6 square array; each cell represents 1 cm².6 × 6 = 366 cm on each side

The 36 equal cells form a 6 by 6 square array; each cell represents 1 cm².

  1. Connect area and side
    Area equals side multiplied by itself because the tile is square.
  2. Count the array
    There are six rows with six unit cells in each row, giving 36 unit cells.
  3. Check the root
    6 × 6 = 36, so the positive side length is 6 cm.
Answer: √36 = 6, so each side is 6 cm.
Check: A 6 cm by 6 cm square has area 36 cm².

Common mistakes and how to avoid them

Writing 36 cm for the side because the area is 36 cm².
Correction: Area and length use different units. Find the number whose square is 36 and use centimetres for the side.

Lesson summary

  • A perfect square has an equal-row square array.
  • A square root gives a nonnegative side length when area is known.
  • Check by squaring the proposed side.

Check your understanding

Question 1

A square has area 49 cm². What is its side length?
  1. 7 cm
  2. 14 cm
  3. 24.5 cm
  4. 49 cm
Show answer and explanation
7 cm
Seven times seven is 49, so the positive side is 7 cm.

Key terms

Perfect square
A whole number made by multiplying a whole number by itself.
Square root
The nonnegative number whose square equals the given number.

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Published by DoAssignment. This reviewed lesson follows Ontario Grade 7 Mathematics, expectation B1.2. It is a study resource, not an official curriculum publication.

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