S.A. and cylinders

Area  of top and bottom  = 2[ pi*R^2  - pi*r^2]  =  2pi[ R^2  - r^2]  = 2pi(R + r) (R - r)

Area  of inside of smaller cylinder  = 2*pi*r*h

Area of outside of larger cylinder = 2*pi*R*h

Total Lateral (side) surface area  = 2*pi*h*[R + r]

Total surface area = 2*pi(R + r) (R - r) + 2*pi*h*[R + r]  = 2*pi*[R + r] [ h + R - r] =

2*pi * [ (6 + 3)mm] [(18 + 6 - 3) mm]  = 1187.52 mm^2

This figure has an INSIDE and an OUTSIDE area AND a TOP and a BOTTOM    ....   First let's find the INSIDE surface area.

pi x d  = circumference = pi x 2(3)     this CIRCUMFERENCE x HEIGHT = area

so for the inside:   Area  is   pi x 2(3) x18 

Now the same thing for the OUTSIDE   pi x 2(6) x 18

Add these together    SO far   pi x 2(3) x 18    +  pi x 2(6) x 18  = 108pi + 216pi = 324pi

Now the top and bottom Area = pi r^2      Subtract the smaller from the larger

   pi (6)^2  - pi (3)^2 = pi 36 - pi 9  = pi 27  = 27pi  (there are TWO of these)  or   54pi

Now add it all together   54pi  +  324pi =  376 pi  = 376 (3.14....) = 1181.24 sq mm (rounded)

Find the surface area of the figure. Round your answer to the nearest hundredth.

H; 18mm

R; 6mm

r; 3mm

\(\begin{array}{lrcll} \text{Area inside is a cylinder with radius r } & A_i &=& ( 2\pi r ) \cdot h \\ \text{Area outside is a cylinder with radius R } & A_o &=& ( 2\pi R ) \cdot h \\ \text{Area top is a ring } & A_{r_1} &=& \pi R^2 - \pi r^2 \\ \text{Area bottom is a ring } & A_{r_2} &=& \pi R^2 - \pi r^2 \\ \hline \\ \text{Area of the figure is the sum } \\ \end{array}\\ \begin{array}{lrcll} & A &=& A_i + A_o + A_{r_1} + A_{r_2} \\ & A &=& ( 2\pi r ) \cdot h + ( 2\pi R ) \cdot h + \pi R^2 - \pi r^2 + \pi R^2 - \pi r^2 \\ & A &=& ( 2\pi r ) \cdot h + ( 2\pi R ) \cdot h + 2\pi R^2 - 2\pi r^2 \\ & A &=& 2\pi ( r \cdot h + R \cdot h + R^2 - r^2 ) \\ & A &=& 2\pi [ h(r + R) + R^2 - r^2 ] \qquad & | \qquad R^2 - r^2 = ( R + r )(R - r)\\ & A &=& 2\pi [ h(r + R) + ( R + r )(R - r) ] \\ & A &=& 2\pi [ h(r + R) + ( r + R )(R - r) ] \\ & A &=& 2\pi (r + R)( h + R - r ) \\\\ & \mathbf{ A }& \mathbf{=} & \mathbf{ 2\cdot (r + R)\cdot ( h + R - r )\cdot \pi } \\ \end{array}\)

\(\begin{array}{lrcll} & A & = & 2\cdot (r + R)\cdot ( h + R - r )\cdot \pi \qquad r = 3\ mm \qquad R = 6\ mm \qquad h = 18\ mm \\ & A & = & 2\cdot (3 + 6)\cdot ( 18 + 6 - 3 )\cdot \pi \\ & A & = & 2\cdot 9\cdot 21 \cdot \pi \ mm^2 \\ & A & = & 378\cdot \pi \ mm^2 \\ & A & = & 1187.52202306 \ mm^2 \\ & \mathbf{A} & \mathbf{ = } & \mathbf{1187.52 \ mm^2 \qquad (\text{ rounded to the nearest hundredth}) }\\ \end{array}\)

or

\(\begin{array}{lrcll} & A & = & 1187.52202306 \ mm^2 \cdot \frac{1\ cm}{10\ mm} \cdot \frac{1\ cm}{10\ mm} \\ & A & = & \frac{1187.52202306}{100} \ cm^2 \\ & A & = & 11.8752202306 \ cm^2 \\ & A & = & 11.8752 \ cm^2 \\ \end{array}\)