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Geometry/Surface Area of a Sphere as a Function of Radius

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Question
No This is not homework.

We want to calculate the surface area of a sphere as it decreases down towards the center of the sphere.  Multiplying surface area by the radius would given an incorrect result as that would not encompass the surface area getting smaller.

Multiplying the full surface area by what  "percent of the radius"  would give a correct value ?

Thanks so much,
Anne

Answer
Hey Anne,

How do you initially calculate the surface area of the sphere?  The surface area of a sphere of radius r is given by A(r)=4πr.  Hence the area decreases quadratically with the radius.  So if the radius of the sphere decreases (or increases, even) to a fraction x of its original value, then we will have:

A(xr)=4π(xr)=x4πr=xA(r)

I hope this answers your question.

Best of luck!
Azeem

Geometry

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Azeem Hussain

Expertise

I welcome your questions on algebra, 2D and 3D geometry, parabolic functions and conic sections, and any other mathematical queries you may have.

Experience

4 years as a drop-in and by-appointment tutor at Champlain College. Private tutor for dozens of clients over the past 8 years.

Publications
CALPHAD: Computer Coupling of Phase Diagrams and Thermochemistry

Education/Credentials
Bachelor of Science, Major Mathematics and Major Economics, McGill University, 2014. Diploma of Collegiate Studies; Pure and Applied Science, Champlain College Saint-Lambert, 2010.

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