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Surface Area of a Right Cylinder

Date: 08/21/2001 at 21:49:10
From: Ramonda Dugas
Subject: Surface area of a right cylinder in centimeters but wanting 
answer in meters

I'm a home school mom trying to learn geometry before I teach it to 
my kids. The problem in my book asked me to find the surface area of 
a right cylinder in centimeters with the dimensions given in meters.

The radius of the end is 10 meters and the length of the cylinder is 
15 meters. I know my formula is 2(area end)+ (CxL).

This is how they achieved the answer:

2(3.14)(10^2)+ 20(3.14)(15) = 628+942=1570(100)(100)= 15,700,000cm^2

What I want to know is why you can't multiply the 10 and 15 by 100 
before you start working the problem to get the correct answer.  It 
does not come out to the same answer doing it that way.

Ramonda Dugas

Date: 08/21/2001 at 23:27:12
From: Doctor Peterson
Subject: Re: Surface area of a right cylinder in centimeters but 
wanting answer in meters

Hi, Ramonda.

There are three ways to do this, all of which are instructive. You 
should get the same answer using either of your two methods.

First, you can convert to the new units at the start as you suggest; 
you will have a radius of 1000 cm and a length of 1500 cm, giving

 A = 2 pi (1000)^2 + pi 2(1000)(1500)
   = 6,280,000 + 9,420,000 = 15,700,000 cm^2

Second, you can find the area and then convert the square units at the 

 A = 2 pi (10)^2 + pi 2(10)(15) = 628 + 942 = 1570 m^2
 1570 m^2 (100)(100) = 15,700,000 cm^2

Third, you can do it all at once, including the conversion factors in 
your formula:

 A = 2 pi (10 m)^2 + pi 2(10 m)(15 m)
   = 2 pi (10 m * 100 cm/m)^2 + pi 2(10 m * 100 cm/m)(15 m * 100 cm/m)
   = 2 pi (1000 cm)^2 + pi 2(1000 cm)(1500 cm)
   = 6,280,000 cm^2 + 9,420,000 cm^2 = 15,700,000 cm^2

Notice that in each term we are multiplying by 100 twice, once for 
each dimension, which explains why we convert from square meters to 
square centimeters by multiplying by the square of 100. This method is 
just the same as either of the others, with labels included to clarify 
what we are doing.

I suspect you may have forgotten to multiply the 10 by 100 before 
squaring it, and only multiplied the first term by 100 rather than 
10000, or something like that. But your suggested method is in fact 
good, and is what I recommend to beginners who haven't yet caught on 
to the conversion of square units.

- Doctor Peterson, The Math Forum   

Date: 08/22/2001 at 10:42:55
From: Ramonda Dugas
Subject: Re: Surface area of a right cylinder in centimeters but 
wanting answer in meters

Thank you.  I thought I should be able to do that at the beginning but
somehow I kept coming up with a higher number.  Must have punched in 
too many zeros in my multiplications.

Associated Topics:
High School Geometry
High School Higher-Dimensional Geometry

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