I need help with a maths question....?

Short of "borrowing" a penguin from the nearest zoo to dip in a tub of water so you can measure the volume of water that spills out, you might model the penguin as a prolate spheroid (ellipse) and find measurements for height, h, from some convenient reference {http://www.brighthubeducation.com/lesson-plans-gra... }. Find a picture of a penguin and measure the width to height ratio, r, and then the volume and area formulas are:

     Volume = π h³ r² / 6

     Area = π h² [ 1/2 + (r²/2) Sin⁻¹√(1-r²) / √(1-r²) ]

Then you can find the surface area to volume ratio, ignoring the bits that stick out, like feet and flippers and the bill. If you care about those bits, when spread out from the body, they will add a lot to the area and but little to the volume, signicantly inreasing the area to volume ratio. Is that important to your application?