Problem
Surface area of a drilled cylinder
A narrower cylinder is bored right through a cylinder along its axis. What is left looks like a thick pipe. Each end is a ring and the hole has a wall of its own. Find the surface area.
Reasoning · Grade 6-2 Shortest Distance and Surface Area
Your answer
Watch out Boring seems to remove area, but the inner wall adds more than it takes.
How to solve
Strategy Identify Subproblems — There is no single formula for a drilled cylinder, but there is a formula for every piece of it. So I walk a finger over the whole outside of the solid and split it into parts that never overlap and leave nothing out: the two flat rings at the ends, the curved wall on the outside, and the curved wall inside the tunnel. Each piece is then a one-line calculation — a ring is a big circle take away a small circle, and a curved wall unrolls flat into a rectangle — and the answer is the sum. Before any of that I have to read the figure correctly, because the two marks on the end face are stacked along one line out from the centre and it is easy to mistake the second one for the radius of the whole face.
1STEP 1
Read the two radii off the end face
The two radii are 1 cm and 3 cm.
r = 1 cm, R = 1 + 2 = 3 cm
A radius is always measured from the centre, so two marks laid end to end along one line out from the centre have to be added. The picture agrees: the end face is drawn three times as wide as the hole.
7.G.B.4Draw A Diagram2STEP 2
Split the surface into four pieces
The surface is two rings, an outer wall and an inner wall.
surface = 2 × (ring) + (outer wall) + (wall of the hole)
Surface area is just the total of every face you could paint, so the safe method is always to name the faces one at a time before working out any areas.
7.G.B.6Identify Subproblems3STEP 3
Area of the two end rings
The two rings total 48 cm².
(3 × 3 × 3) - (1 × 1 × 3) = 27 - 3 = 24, 24 × 2 = 48 cm²
A ring has no formula of its own, but it is exactly what is left after a small circle is taken from a big one, so one subtraction handles a shape that looks awkward.
7.G.B.4Change Focus Count The Complement4STEP 4
The outside wall unrolls into a rectangle
The outer wall unrolls to 144 cm².
(6 × 3) × 8 = 18 × 8 = 144 cm²
Unrolling turns a curved surface into a plain rectangle, and the area of a rectangle is something a young solver already owns; the only new fact needed is that the width of the unrolled sheet is the circumference.
6.G.A.4Create A Physical RepresentationThe outside wall unrolls into a plain rectangle whose width is the circle's rim and whose height is the cylinder's height.
Why?
Unrolling bends the wall flat without stretching it, so every length on the wall is carried across exactly.
🧱Rigid motion preserves length and angleSliding, turning, or flipping a shape lays it exactly onto a copy, so every length and angle stays the same (folding is the flip case).
Why?
The whole surface is the two end rings and the two walls put together, with no gap and no overlap between them.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
5STEP 5
The wall of the hole unrolls too
The hole's wall is 48 cm².
(2 × 3) × 8 = 6 × 8 = 48 cm²
It is tempting to think drilling only takes surface away, but the tunnel wall is skin that did not exist before, and it is counted exactly like the outside wall.
6.G.A.4Create A Physical Representation6STEP 6
Add the four pieces
Adding them gives 240 cm².
48 + 144 + 48 = 240 cm²
Once the surface has been cut into pieces that do not overlap, the only thing left to do is add — the hard thinking was all in the splitting.
7.G.B.6Identify Subproblems7STEP 7
Check it by starting from the whole cylinder instead
Starting from the whole cylinder also gives 240 cm².
198 - 6 + 48 = 240 cm²
Drilling both removes skin and creates skin, and seeing that the new tunnel wall (48) is far bigger than the two lost discs (6) explains why a drilled cylinder has more surface than a solid one, not less.
7.G.B.6Change Focus Count The ComplementAnswer
240 cm²
48 + 144 + 48 = 240
The units are right: every piece was a length times a length, so the total is in square centimetres, and the question asks for square centimetres. The size is right too. The outside wall alone is 144, and the two end rings and the tunnel wall are each 48, so 240 is only a little over one and a half times the outside wall — sensible for a shape that is mostly outside wall. It also sits just above the 198 of the un-drilled cylinder, which is what has to happen: drilling loses two small discs worth 6 but gains a tunnel wall worth 48. A quick sanity test on the misreading this problem invites: if the end face had radius 2 cm instead of 3 cm the answer would come out as 2 x (12 - 3) + 12 x 8 + 6 x 8 = 18 + 96 + 48 = 162, not 240, so the picture and the answer agree that the radius is 1 + 2 = 3 cm.
Takeaway
Drilling a hole right through takes away two little discs but adds a whole tunnel wall, so name every surface you could paint before you add anything up.
- Read the two radii off the end face
- Split the surface into four pieces
- Area of the two end rings
- The outside wall unrolls into a rectangle
- The wall of the hole unrolls too
- Add the four pieces
- Check it by starting from the whole cylinder instead