Depth · one concept from first principles · step 2 of 4
One full roll moves it its own circumference
A hoop with a radius of is rolled in one direction and makes and a half turns. How far does it travel, in ?
Geometry
Your answer
How to solve
Strategy Draw a Diagram — Rolling lays the hoop's rim out along the ground, so one turn covers one circumference. Find that length once and the number of turns does the rest.
1STEP 1
Find how far one turn goes
One turn lays the whole rim along the ground, so it covers the circumference: radius times two times 3.14.
30 × 2 × 3.14 = 188.4
One turn is 188.4 cm.
7.G.B.4Draw A DiagramOne full turn moves the hoop forward by exactly its circumference.
Why?
Rolling without slipping lays each point of the rim onto one point of the ground and never onto two, so the rim and the track it leaves match point for point.
🧱One-to-one correspondenceIf each thing here pairs with exactly one thing there, the two groups are the same size.
Why?
Unrolling the rim does not stretch it, so the track is exactly as long as the rim itself.
🧱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
Multiply by the number of turns
Four and a half turns cover four and a half circumferences.
188.4 × 4.5 = 847.8
The hoop travels 847.8 cm.
7.G.B.4Identify SubproblemsAnswer
847.8 cm
Split it up: four turns are 753.6 cm and the half turn is 94.2 cm, which come to 847.8 cm.
Takeaway
One turn of a wheel covers exactly one circumference.
- Find how far one turn goes
- Multiply by the number of turns
Where next?
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