Piano Soundboard shape and size

The size of the soundboard is determined by the size of the piano. They make different size pianos for different venues, home concert, etc.

Basically here are some design reasons why:
1. The panel is not graded, and or is too thick. The Steinway diaphramatic method goes too far as a grading method and often leaves the panel structurally too thin and weak.
2. The rib structure is too stiff. Most rib designs in modern pianos today have way too much mass. I have measured ribs that are 1-1/4" tall in some cases. Early American pianos often stayed around 3/4" to maintain strength and flexibility. Another factor is wood choice. Steinway uses sugar pine ribs, and others will use Spruce ribs. Spruce is stiffer than pine.

Sound-board's size is proportional to the scale design of the piano. It is worth understanding that not the whole soundboard area is functional, and in certain cases a smaller soundboard than average may provide tonal improvements.
The specifications of the soundboard-system will determine the acoustic properties of the soundboard, that ultimately affects how energy flows through the board while the energy is radiated and consumed as well. [copy and paste] (I won't say from where for now ;)) magnitude of the impedance (Z), and its phase angle (?), and the sound level (L). These three curves summarize the acoustic properties of the soundboard.
However without trial an error we can take measurements to create specifications to manufacture a soundboard.

Ribs are fundamental contributors to the performance and acoustic properties of the soundboard: [copy and paste] A stiffening of the soundboard would improve the sound radiation efficiency, as a stiff soundboard is less inclined to subdivide into small vibrating areas. One purpose of the ribs is precisely to stiffen the soundboard.
It was important to provide this explanation as it supports the fact that the performance of the sound-board is not simply connected to size/shape.
The topic of a piano soundboard can generate technically hundreds of pages, if a scientific explanation is required, yet I believe that work-experience and the human ear lead also to very satisfactory results as well.

Understanding soundboard response is very complex. It is too large to act like a piston, which, by the way is how speakers ideally behave. Additionally, the very stiff connection between the soundboard and the rim also inhibit its motion. Also, of course, a soundboard is not isotropic--its stiffness is different in different directions and different positions. Motion cannot be transmitted instantaneously--it travels typically in transverse or compression waves. So, when string movement is transmitted to the bridge it radiates outward with finite velocity until it is damped out or reflects off the rim. A soundboard has a fundamental (lowest) resonance. Below this frequency its ability to radiate sound becomes increasingly reduced (by the square of the ratio between the resonance and driving frequency). Honestly, to sort this out analytically seems impossible. Finite element analysis seems the best way to model and understand soundboard parameters. This is not a problem for large companies, though smaller ones likely wouldn't want to purchase the pricey software, and wouldn't want to pay a consultant to do the analysis.