This paper discusses an investigative approach to problems concerned with approximations. Two problems are analysed in detail and both are inspired by their historical origins. The first examines an ancient Chinese formula for calculating the area of a segment. The second looks at Durer's construction for the trisection of an angle. It is argued that historical problems can provide a rich source for such an investigative approach. Discussion of the second problem also raises fundamental issues about the nature of "conceptual tools" and "conceptually exact solutions" to geometric construction problems, particularly in the light of using dynamic geometry environments such as Sketchpad.