The length √2

The idea

The hypotenuse c of an isosceles right triangle with equal sides 1 unit has a square of area 2 sq. units on it, so c² = 2 and c = √2 units.

How long is the diagonal?

A square has side 1 unit. Its diagonal cuts it into two isosceles right triangles. How long is the diagonal?

We know the square built on this diagonal has area 2. The diagonal is the side of that square.

Which length do you think the diagonal is?

• exactly 1 unit
• exactly 1.5 units
• exactly 2 units

Swing the diagonal

The diagonal is like a compass arm. Fix one end at 0 and swing the other end down onto the number line. The amber square is built on the diagonal.

A length with a new name

The hypotenuse c of an isosceles right triangle with equal sides 1 unit has a square of area 2 sq. units on it. c × c = c² = 2 So c = √2 units.

√2 is read “root two”. It is the number that gives 2 when multiplied by itself.

1 × 1 = 1 is too small and 2 × 2 = 4 is too big. So √2 lies between 1 and 2.

Notes

A square of side 1 has a diagonal of length c. The square on it has area 2, so c × c = 2, c = √2. It lies between 1 and 2 on the number line.

Check yourself

The diagonal of a square of side 1 is c. Which equation is true?

Where does √2 lie on the number line?

The diagonal of a unit square is √2 units. What is √2 × √2?

Answer: 2

c × c = 2 and c = √2, so √2 × √2 = 2.

A square has area 2 sq. cm. How long is its side?

• c × c = 1. That is the area of the small square (1 × 1). The square on the diagonal has area 2.
• c + c = 2. That would make c = 1, the side of the small square. The diagonal is longer.
• c × c = 2 — correct. Yes! The square on the diagonal has area 2, and its side is c.
• Between 0 and 1. 1 × 1 = 1 is less than 2, so √2 is bigger than 1.
• Between 1 and 2 — correct. Yes! 1 × 1 = 1 is too small and 2 × 2 = 4 is too big.
• Between 2 and 3. 2 × 2 = 4 is already bigger than 2, so √2 is smaller than 2.
• √2 cm — correct. Yes! side × side = 2, so the side is √2 cm.
• 2 cm. A square of side 2 cm has area 2 × 2 = 4 sq. cm.
• 4 cm. A square of side 4 cm has area 16 sq. cm. The side must be smaller than 2.