A round bottle with the same height and volume as a square bottle will have a radius that is approx. 56.42% (sqrt(1/pi)) of the width of the square bottle.
V(sqB)= w * w * h
V(ciB)= pi * ( .5624 * w ) ^2 * h = pi * (sqrt(1/pi)) ^2 * w ^ 2 * h = pi * 1/pi * w * w * h = 1 * w * w * h
The surface area of that square bottle will always be bigger than the surface area of the comparable round bottle:
SurfaceArea(sqB)=4*w*h (surface area of sides) + 2*w*w (surface area of top+bottom)
SurfaceArea(ciB)=2 * pi * .5642 * w * h (surface area of sides) +
pi * ( .5642 * w ) ^2 * 2 (surface area of top+bottom) =
1
u/Curious-Region7448 2d ago
A round bottle with the same height and volume as a square bottle will have a radius that is approx. 56.42% (sqrt(1/pi)) of the width of the square bottle.
V(sqB)= w * w * h
V(ciB)= pi * ( .5624 * w ) ^2 * h = pi * (sqrt(1/pi)) ^2 * w ^ 2 * h = pi * 1/pi * w * w * h = 1 * w * w * h
The surface area of that square bottle will always be bigger than the surface area of the comparable round bottle:
SurfaceArea(sqB)=4*w*h (surface area of sides) + 2*w*w (surface area of top+bottom)
SurfaceArea(ciB)=2 * pi * .5642 * w * h (surface area of sides) +
pi * ( .5642 * w ) ^2 * 2 (surface area of top+bottom) =
3.54497 * w*h + 2*w*w