Square Numbers and Square Roots

What is a square number?

To square a number means to multiply it by itself. We write it with a small raised 2, called an exponent or power. So:

5² = 5 × 5 = 25

We read 5² as "five squared" or "five to the power of two". A square number (also called a perfect square) is the result you get when you square a whole number. The first few square numbers are:

1, 4, 9, 16, 25, 36, 49, 64, 81, 100

Be careful: squaring is not doubling. 5² is 5 × 5 = 25, not 5 + 5 = 10. This idea is part of the bigger topic of exponents and powers, where the power can be any number, not just 2.

Why is it called "square"?

The name comes from geometry. Imagine a square made of small unit squares. If each side is 4 units long, the square is made up of 4 × 4 = 16 small squares. The total number of squares is the side length squared — which is exactly why we call it "4 squared". This connects squaring to area: the area of a square is side × side.

A table of square numbers

It is worth knowing the first set of squares by heart.

Number (n)	n² (n squared)	Calculation
1	1	1 × 1
2	4	2 × 2
3	9	3 × 3
4	16	4 × 4
5	25	5 × 5
6	36	6 × 6
7	49	7 × 7
8	64	8 × 8
9	81	9 × 9
10	100	10 × 10
11	121	11 × 11
12	144	12 × 12

If you already know your times tables, you know most of these — they are just the "doubles" along the diagonal of the multiplication grid.

Squaring bigger numbers

You can square any number, not just the small ones.

Worked example: Find 15².

1. 15² = 15 × 15.
2. Break it up: 15 × 15 = 15 × 10 + 15 × 5 = 150 + 75.
3. Add: 150 + 75 = 225.
4. So 15² = 225.

Worked example: Find 20².

1. 20² = 20 × 20 = 400.

What is a square root?

A square root does the opposite of squaring. It asks: "What number, multiplied by itself, gives this?" The symbol is √, called the radical sign.

√25 = 5 because 5 × 5 = 25.

So squaring and square-rooting undo each other, just like adding and subtracting are opposites. If you square 6 to get 36, taking the square root of 36 brings you back to 6.

Worked example: Find √81.

1. Ask: what number times itself gives 81?
2. Test: 9 × 9 = 81.
3. So √81 = 9.

Worked example: Find √144.

1. What times itself gives 144?
2. 12 × 12 = 144.
3. So √144 = 12.

Knowing your square-numbers table makes finding square roots quick — you just read the table backwards.

Estimating square roots that are not whole

Most numbers are not perfect squares, so their square roots are not whole numbers. For example, √20 is not a tidy value, but we can estimate it by trapping it between two perfect squares.

Worked example: Estimate √20.

1. Find the perfect squares just below and above 20. The square below is 16 = 4² and the square above is 25 = 5².
2. Since 20 is between 16 and 25, √20 is between 4 and 5.
3. Because 20 is closer to 16 than to 25, the answer is a bit above 4 — about 4.4.
4. Check: 4.4 × 4.4 = 19.36, close to 20. (A calculator gives √20 ≈ 4.47.)

Worked example: Estimate √50.

1. 49 = 7² and 64 = 8², so √50 is between 7 and 8.
2. 50 is very close to 49, so √50 is just above 7 — about 7.1.
3. Check: 7.1 × 7.1 = 50.41, very close to 50.

Where squares and roots are used

Squares and square roots appear all over the place:

• Area. Finding the area of a square uses squaring; finding a side length from an area uses a square root. If a square room has an area of 36 m², each side is √36 = 6 m. This links to area and perimeter.
• The Pythagorean theorem. Working out distances in right-angled triangles relies on squares and square roots.
• Real life. Designers, builders, and game programmers use them constantly — for example, to work out the size of a screen or the distance between two points.

Practice activity

1. Find 7², 9², and 12².
2. Find √36, √64, and √100.
3. Is 30 a perfect square? Why or why not?
4. Between which two whole numbers does √40 lie? Give a one-decimal estimate.

Answers:

1. 7² = 49, 9² = 81, 12² = 144.
2. √36 = 6, √64 = 8, √100 = 10.
3. No. There is no whole number that multiplies by itself to give 30 (5² = 25 and 6² = 36, with nothing in between), so 30 is not a perfect square.
4. 36 = 6² and 49 = 7², so √40 is between 6 and 7. Since 40 is close to 36, a good estimate is about 6.3 (check: 6.3 × 6.3 = 39.69).

Summary

Squaring a number means multiplying it by itself (6² = 36), and the result is a square number or perfect square. A square root undoes squaring (√36 = 6) by asking which number times itself gives the result. The perfect squares — 1, 4, 9, 16, 25, 36, … — are worth memorising, because knowing them lets you find exact roots instantly and estimate the in-between roots by trapping them between two perfect squares.