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Advanced Math/area of circle inscribed on a sphere

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Expert: - 11/28/2009


Question
I am not a student.  So this is not from a book or a particular study of math.  Just a question a couple of adult friends are pondering.  Imagine a large polka dot on the side of the sphere.  What is the area of that polka dot?

I have a high school math teaching license, and I've had 3 years of calculus, so try me.  

Thank you for your time.

Anne


Answer

Sphere
Questioner: Anne
Country: United States
Category: Advanced Math
Private: No
Subject: area of circle inscribed on a sphere
Question: I am not a student.  So this is not from a book or a particular study of math.  Just a question a couple of adult friends are pondering.  Imagine a large polka dot on the side of the sphere.  What is the area of that polka dot?

I have a high school math teaching license, and I've had 3 years of calculus, so try me.  

Thank you for your time.

Anne
...........................................
OK, Anne, three years might be enough.  Here goes --

I shall assume:

The radius of the SPHERE is r.  This will be  a kind of 'area of revolution' exercise:

I shall write t (= theta) as the upper half of a central angle.

The element of arc on a CIRCLE is  ds = r dt

Rotate this around your radius to get a 'ring element' of area.

This element of area is  dA = 2 pi y ds

Integrate this from  t = 0 to  t = x.

dA = 2 pi y r dt

dA = 2 pi (r sin t) r dt

dA = 2 pi r^2 sin t dt

Integrate for t = 0 to x

A = 2r^2 (-cos t)  from 0 to x

A = 2r^2 [- cos x - (- cos 0)]

A = 2r^2 [- cos x - (- 1)]

A = 2r^2 [- cos x + 1]

A = 2r^2 [1 - cos x]
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How about a little 'check' on this?  

Suppose x = pi, making the whole sphere, that is:

A = 2r^2 [1 - cos pi]

A = 2r^2 [1 + 1] = 4 pi r^2

Yes, that is the formula for surface area of a sphere.

So maybe we are OK.  

Paul Klarreich

Expertise

I can answer questions in basic to advanced algebra (theory of equations, complex numbers), precalculus (functions, graphs, exponential, logarithmic, and trigonometric functions and identities), basic probability, and finite mathematics, including mathematical induction. I can also try (but not guarantee) to answer questions on Abstract Algebra -- groups, rings, etc. and Analysis -- sequences, limits, continuity. I won't understand specialized engineering or business jargon.

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I taught at a two-year college for 25 years, including all subjects from algebra to third-semester calculus.

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