Frequently Asked Questions

How do you use geometry to square root a number?

Begin with a segment equal in length to the number, x and extend the length by one unit. Use this new length as a diameter to produce a semicircle. Construct a perpendicular to the diameter at the unit mark and allow the perpendicular to intersect the semicircle, the length of the new segment will be √x.

We can prove this as follows.

The diameter is x + 1 and as c is equal to the radius, c = ½(x + 1). Hence b = ½(x – 1).

Using the Pythagorean Theorem,

a² = c² – b²
  = ¼(x + 1)² – ¼(x – 1)²
  = ¼(x² + 2x + 1) – ¼(x² – 2x + 1)
  = ¼ × 4x
  = x

Therefore a is equal to √x.