R and r are the radius of two circles (R > r). If the distance between ­the centre of the two circles be d, then length of common tangent of two circles is :

We have , $$\text{Hypotenuse}^{2}$$ = $$\text {base}^{2}$$ + $$\text {perpendicular}^{2}$$
Radii of the circles which intersect the tangents are parallel as both of them are perpendicular to the tangent.
Now, we draw a line parallel to the line which joins the centre of both the circles which intersects the extended radius of small circle at A and let the extended length be ‘a’
So, R = r + a i.e a = R - r
Now a right angled triangle is formed as shown in the figure as tangents and radii intersect at 90°
Applying Pythagoras theorem:
$$\text{(Length of tangent)}^{2}$$ + $$a^{2} = d^{2}$$
$$\text{(Length of tangent)}^{2}$$ = $$d^{2} - (R - r)^{2}$$
Length of tangent = $$\sqrt{d^{2} - (R - r)^{2}}$$