Question

Dear Mitch,

My friend and I love math and are on our school's math team. (We're in seventh grade). We even try to come up with weird problems and try to solve them together sometimes on weekends if we can't think of anything else to do. (Some kids call us 'geeks' but we don't care! Anyway, the problem is this: My older brother is in eleventh grade, and he thinks he's the total king of the math world. He is very, very good in math and is on his school's math team, which is fine, but the problem is that he really tries to annoy me and my friend by telling us that what we're doing is "not bad, but not completely correct either". And now I have a certain particular question that he is so sure about but my friend and I aren't sure he's right (sometimes my brother can be a serious bluffer). Anyway, it's an Easter question that was in the local newspaper, and it asked about finding the surface area of an Easter egg so you could figure out how much paint or whatever to buy if you had to paint 500 eggs for a big school event. My friend and I were using the formula for the surface area of a sphere, figuring that even though eggs are not really perfect spheres like ping-pong balls, if you squeeze a sphere around its middle it would get a little longer and a little narrower and would basically be an egg shape and since you haven't stretched its surface, just reshaped it a little, it would still be the same amount of area all the way around.

My brother said, No, we're very wrong. He said that there's a special formula for eggs, and he said that the name of the shape of an egg is a "spheroid", but that we couldn't learn how to calculate it until we were his age and knew much more math than now. Is that really true?

Sincerely,

Gregg H.

Fort Worth, TX

Answer

Dear Gregg,

I have an idea: Go over to your brother and ask him if he would be more specific. When he says, 'Huh?' You say, 'Well, I was wondering about the spheroid, and what I'm not sure about is whether I should use the formula for a PROLATE SPHEROID or an OBLATE SPHEROID.'

(And "oblate spheroid" is a sphere that's been flattened a little on top and bottom, like an M&M candy or a balloon on which a child is sitting. A "prolate spheroid" is a sphere that's been stretched from the top and bottom, like the shape of one of those little hotdogs or sausages in those 'pigs-in-blankets' at a party.)

Mathematically, the equation that describes spheroids in a three dimensional graph system uses three radii (one for each of the three-dimensions), and when all three radii are equal you have a perfect sphere... But, when only two are equal, you have a spheroid, and the spheroid will be either the oblate kind or the prolate kind.

After he answers you, if he does, thank him for his help. THEN, casually mumble to yourself that you must've been confused, because you thought an egg-shape is neither an oblate spheroid nor a prolate spheroid. You thought it was a shape called a 'scalene ellipsoid', which, like a scalene triangle, has three unequal sides, or, in this case, three different – 'unequal' -- radii. AND THAT IS ACTUALLY THE CORRECT ANSWER.

Why?

Because an egg is NOT stretched or flattened the same amount on two 'sides' but is stretched more one way than the other. WHY? Well, scientists believe that an egg has the shape it does because over the course of evolution (or 'development') this interesting shape does not roll away from the mother who laid it, but will only roll around in circles until the mom reclaims it and resumes sitting on it to keep it warm for its big day!

Hope that helps,

Mitch

P.S. Soon, I will post some very easy problem-solving ways to figure out the surface area of an egg – and you won't have to know any fancy math that you didn't already know when you were back in fifth or sixth grade!