For the mid-1970s, F1 stipulated that the maximum width of the front wing was 1.5 m. Considering the needed room for the driver's feet, the steering mechanism, suspension and the normal front tyre size, this meant the front tyres projected above and out to the sides of the wing. P34's basic concept was to use a tyre that would be small enough to fit entirely behind the wing. This would have two effects; one would be to lower overall drag and thus improve speed on straights and the other was to clean up overall aerodynamics so the rear wing would receive cleaner airflow.

However, given the space limitations such a tyre would have to be quite small, eventually settling on a 10 inches (250 mm) diameter. This had too small a contact patch to offer reasonable cornering performance, which led to the use of four wheels instead of two. Adding more wheels also had the advantage of offering more total brake area. The downside was increased complexity of the steering system and a physically larger suspension system. The steering complexity was solved by connecting only the front pair of wheels to the steering wheel and connecting the rear set to the front with a bell crank. Initially considered to be a problem area, Joel Rosinsky later declared that "The steering is so gentle and absolutely free of reaction that you might have thought it was power-assisted!"

The new design was unveiled at the Heathrow Hotel in late September 1975. The car was initially kept under a tarpaulin with hoops over the wheels that make it look conventional, leading to astonishment when the tarp was pulled off. Some in the audience were convinced the design was a publicity stunt. The car first took to the track at Silverstone on 8 October 1975, and after more tests, Tyrrell decided to build two more examples with a slightly longer wheelbase to race in the 1976 season.