Total surface area of a cylinder

Question

A log with diameter 60 cm is 3 m in length. Its hollow centre is 20 cm in diameter.

Find the surface area of the log in cm2, including the ends and the inside, correct to one decimal place.

Answer 1

It may be easiest to visualize this separately for each portion. A "tube" shape has an outer surface, inner surface, and 2 ends (which normally will be the same)

Working in cm

For the specified shape:

Outer: Simply the area of the outsdie of the larger cylinder, not counting the ends.

This is the circumference (c) times the length (l), C = pi*d, so the area is pi*60*300 = pi*18000

Inner: Simply the inner surface of the smaller cylinder: l is the same, and d is 20, so area = pi*20*300 = p*6000

Ends: Here we use the difference between the larger and smaller end areas.

for the larger: A = pi*d² / 4 = pi*(60)² / 4 = 900*pi

and the smaller: A = pi*(20)² / 4 = 100*pi

difference: 800*pi

The are 2 ends so the total end area is 1600*pi

Add it all up: A(total) = pi ( 18,000 + 6,000 + 1,600) = pi* 25,600 = 80424.7719319

to one decimal place: 80424.8

Answer 2

These are tough. Think about cutting a cylinder from one end to the other and unrolling it. We now have a rectangle and the circumference of the circle end is the length of the board and the height being the width. In this instance, we have a board.

Outside circumference = 60pi

If we unroll it, the area of each side of the board is 60pi(300) = 18,000pi cm²

The top edges are each 60pi(60 - 20) = 2400pi cm²

The side edges are each 300(40) = 12,000 cm²

There are 2 of each side.

2(18,000pi + 2400 + 12,000) = 141,897.3 cm²