The side of a square, and the long side of a triangle

Word	What it means
Square of a number	The number times itself: 6² = 6 × 6 = 36, the area of a square with sides 6 long.
Perfect square	The square of a whole number: 1, 4, 9, 16, 25, 36, … 40 is not one.
Square root, √	The side of the square: √36 = 6, because 6 × 6 = 36.
Factor pair	Two whole numbers that multiply to make a number: 4 × 9 is a factor pair of 36.
Prime factorization	A number written as a product of primes, numbers whose only factors are 1 and themselves: 36 = 2 × 2 × 3 × 3.
Benchmark	A nearby perfect square, used to estimate a square root. For √50 the benchmarks are 49 and 64.
Right triangle, legs, hypotenuse	A right triangle has a 90° angle, marked with a small square. Its legs, a and b, make the right angle; the hypotenuse, c, is across from it and is always the longest side.

6² = 36, so √36 = 6  ·  12² = 144, so √144 = 12  ·  15² = 225, so √225 = 15

Factor pairs show it. Each factor pair of 36 is a rectangle with an area of 36: 1 × 36, 2 × 18, 3 × 12, 4 × 9 and 6 × 6. Only 6 × 6 is a square, and 6, the factor paired with itself, is the square root. The pairs for 40 are 1 × 40, 2 × 20, 4 × 10 and 5 × 8. None is a square, so 40 is not a perfect square.

Prime factors, for bigger numbers. A perfect square's prime factors split into two identical groups: 324 = 2 × 2 × 3 × 3 × 3 × 3 = (2 × 3 × 3) × (2 × 3 × 3) = 18 × 18, so √324 = 18. But 72 = 2 × 2 × 2 × 3 × 3 has three 2s, which cannot be split evenly, so 72 is not a perfect square.

Worked example. A square garden has an area of 196 m². How much fencing goes around it? Each side is √196 = 14 m, because 14 × 14 = 196, so four sides need 4 × 14 = 56 m. Dividing the area by 4 instead gives 49, which mixes up area and perimeter.

Worked example: √50.

1. Benchmarks: 49 = 7² and 64 = 8², so √50 is between 7 and 8.
2. Closer to which? 50 is 1 more than 49 but 14 less than 64, so √50 is much closer to 7: about 7.1.
3. Check with a calculator: √50 = 7.0710678…, or 7.07 to two decimal places.

A calculator is only close. The digits of √50 go on forever and the screen shows only the first few, which is why 7.07 × 7.07 = 49.9849, not 50.

Nearly halfway. √90 is between √81 = 9 and √100 = 10. 90 is 9 above 81 and 10 below 100, so √90 is just under halfway: about 9.5. (It is 9.49.)

Backwards. Which whole numbers have a square root between 6 and 7? The ones between 6² = 36 and 7² = 49: 37 to 48.

a² + b² = c², where c is the hypotenuse

Worked example: the hypotenuse. The legs of a right triangle are 9 cm and 12 cm.

1. Square the legs: 9² = 81 and 12² = 144.
2. Add: c² = 81 + 144 = 225.
3. Take the square root: c = √225 = 15 cm.

Adding the legs gives 9 + 12 = 21 cm, the two legs laid end to end. The straight way across is always shorter.

Worked example: a leg. A 5 m ladder leans against a wall with its foot 2 m out. How high up the wall does it reach?

1. Find the hypotenuse first. The right angle is where the wall meets the ground, and the ladder is across from it, so c = 5.
2. Subtract the squares: 5² − 2² = 25 − 4 = 21.
3. Take the square root. √21 is between 4 and 5, a little closer to 5. A calculator gives 4.58, so the ladder reaches about 4.6 m.

Is it a right triangle? Run the theorem backwards. Sides 8, 15 and 17: 8² + 15² = 64 + 225 = 289 = 17², so yes. Sides 7, 9 and 12: 7² + 9² = 49 + 81 = 130, but 12² = 144, so no. Builders check a corner this way: 3 m along one wall, 4 m along the other, and exactly 5 m between the marks.

• Halving instead of rooting. √50 is about 7.1, not 25.
• Adding the sides instead of their squares. Legs of 9 and 12 give a hypotenuse of 15, not 21.
• Adding when finding a leg. A leg is shorter than the hypotenuse, so subtract: 5² − 2², not 5² + 2².
• Stopping at c². c² = 225 means c = 15. The last step is the square root.
• Using the theorem without a right angle. It only works on right triangles.