Depth · one concept from first principles · step 2 of 4
Find the matching lengths in a cylinder net
The figure at the right is the net of a cylinder whose base has a circumference of . What is the perimeter of this net, in ?
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — The one length that has to be worked out is the rectangle's width, and the net itself gives it: rolled up, that side becomes the rim of the base, so it is the base's circumference. Everything else is already known.
1STEP 1
Match the rectangle's width to the base
Rolling the net up wraps that side exactly once round the base, so the two lengths are the same.
The rectangle is 9.42 cm wide.
6.G.A.4Visualize Spatial RelationshipsThe rectangle's width is the base's circumference.
Why?
Rolling the rectangle up wraps that side exactly once round the base, so each point of the width meets one point of the rim.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Rolling bends the side without stretching it, so its length is the rim's length.
🧱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).
2STEP 2
Measure the rectangle's outline
Two widths and two heights.
9.42 × 2 = 18.84, 5 × 2 = 10
The rectangle's outline is 28.84 cm.
7.G.B.4Draw A Diagram3STEP 3
Add the two circles
Each circle touches the rectangle at only one point, so its whole circumference is on the outline.
9.42 × 2 = 18.84
The circles add 18.84 cm.
7.G.B.4Draw A Diagram4STEP 4
Add it all up
The outline is the rectangle's four sides plus the two circles.
28.84 + 18.84 = 47.68
The net's perimeter is 47.68 cm.
7.G.B.6Identify SubproblemsAnswer
47.68 cm
Count the 9.42s: the rectangle's top and bottom are one each and the two circles are one each, so four of them, plus the two 5 cm sides -- 9.42 × 4 = 37.68 and 37.68 + 10 = 47.68.
Takeaway
In a net, the side that rolls into a rim is exactly as long as that rim.
- Match the rectangle's width to the base
- Measure the rectangle's outline
- Add the two circles
- Add it all up
Where next?
Another one like thissuggested
Same look, different logicsuggested
Step back and solidifysuggested
A parent dashboard for the family lives at sensimlab.com.
The same shape shows up later on the AMC, as Net Folding / Face Adjacency. Why the shapes repeat →