Suppose you have one of those puffy helium balloons. How do you figure the area and the volumn?

Answer 1 is good . It was good enough for Archimedes . Once you know the volume displaced , you can just use your volume of a sphere formula to work out the surface area of your balloon. Even though it's a heart shape it will have to accommodate the same volume and as such the shape change won't alter the surface area required- only its form.

Well you could make a reasonable mathematical model of it, such as that a heart shape is two semicircles joined at 90 degrees, and with their opposite ends extended to make two sides of a right triangle. Then a three-dimensional heart has elliptical cross-sections in planes aligned with the vertical direction of the heart, with major axis equal to the height from bottom edge to top edge and all equal in eccentricity. Then integrate to your heart's delight.

Is this a 'home made' question? Why would you need its area?

If you did manage to work out a volume, an area is easy.

Too hard to measure volume, or methods too inaccurate.

put the ballon in a deep tub, fixed. then fill the tub with water, mark where the water level is, then remove the ballon, mark where the new water level is then you can calculate the volume. eurika.