The Pythagorean Theorem · Grade 8

What is a square number and what is a square root?

Squaring a number means multiplying it by itself, and taking a square root is the opposite operation — it gives back the number the square came from. Seven squared is forty-nine, so the square root of forty-nine is seven. These two operations are everything Pythagoras' theorem is made of.

Learning objectives

Two operations that undo each other

Squaring a number means multiplying it by itself. Five squared is five times five, which is twenty-five. The name comes from the shape: a square whose side is five units contains exactly twenty-five unit squares.

A square root runs the other way. It asks which number, multiplied by itself, gives what sits under the root sign. If a square has an area of twenty-five, its side is five — and that is precisely what the root returns.

Because they undo each other, the root of a number squared is the number itself. That is what will later let you pull a side length out of an equation where it appears squared.

When the root is not a whole number

Only a few numbers are perfect squares: 1, 4, 9, 16, 25 and so on. For everything else the root is not whole, and it neither terminates nor repeats.

You can still say which two whole numbers it falls between. The root of 50 lies between 7 and 8, because 49 is 7 squared and 64 is 8 squared — and it is much closer to 7, since 50 is much closer to 49.

That estimate is worth more than it looks. It catches almost any mistyped calculator entry, and it lets you check an answer without redoing the whole calculation.

When to round and when not to

If the question asks for an exact answer, leave the root sign in place. If it asks for a length in centimetres, round — usually to two decimal places, unless told otherwise.

Rounding in the middle of a calculation rather than at the end is a common source of error. Each rounding adds a small drift, and multiplying carries that drift into the first digit of the answer. Round once, at the end.

Worked examples

  1. Find 12 squared and the square root of 144

    1. 12 squared means 12 times 12
    2. 12 times 12 is 144
    3. The root is the opposite operation, so the root of 144 gives back 12

    Answer: 12 squared is 144, and the root of 144 is 12

  2. Between which two whole numbers does the square root of 90 lie?

    1. 9 squared is 81, which is less than 90
    2. 10 squared is 100, which is more than 90
    3. So the root lies between 9 and 10
    4. 90 is nearer 81 than 100, so the root is nearer 9

    Answer: Between 9 and 10, closer to 9. On a calculator: about 9.49

  3. A square has an area of 64 square centimetres. How long is its side?

    1. The area of a square is the side length squared
    2. So the side is the square root of the area
    3. The root of 64 is 8, because 8 times 8 is 64

    Answer: The side is 8 centimetres

Common mistakes

Multiplying by two instead of by itself
Seven squared is forty-nine, not fourteen. The exponent says how many times the number appears in the product, not what to multiply it by. This single error ruins more Pythagoras questions than everything else combined.
Taking the root of each term inside a sum
The root of a sum is not the sum of the roots. The root of nine plus sixteen is the root of twenty-five, which is five — not three plus four. One numerical check shows this immediately.
Rounding partway through
Every rounding introduces drift, and multiplication magnifies it. Keep the full value on the calculator until the last step and round only the final answer.

What to remember

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