Problem
Match the traced circle's circumference to the cone
A cone lying on its side is rolled about its apex and comes back to where it started after a known number of turns. We need its base radius.
Geometry
Your answer
How to solve
Strategy Visualize Spatial Relationships — Two circles are involved and the same length gets used twice. The path is one big circle; the cone's own rim rolls along it. Five of the small rims must lay out the whole big circle, and that one sentence is the equation.
1STEP 1
Find the circle the cone traces
The apex stays fixed and the point where the base touches the ground stays a slant height away from it, so it travels round a circle of radius 10 cm.
10 × 2 × 3.14 = 62.8
The path is 62.8 cm round.
7.G.A.1Visualize Spatial RelationshipsThe cone's base traces a circle whose radius is the slant height.
Why?
The apex stays put while the cone rolls, and the point where the base touches the ground stays one slant height away from it the whole time.
🧱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?
A point held at a fixed distance from a fixed centre sweeps out a circle of exactly that radius.
🧱A circle's radius is constantEvery point on a circle sits exactly one radius from the center, so all radii of a circle are equal.
2STEP 2
Say what five turns cover
Each turn of the cone lays its own base circumference along the path, and five turns bring it right back to the start.
62.8 ÷ 5 = 12.56
The base's circumference is 12.56 cm.
7.G.B.4Convert To Algebra3STEP 3
Recover the radius
Divide the circumference back out by 2 and by 3.14.
12.56 ÷ 3.14 = 4, 4 ÷ 2 = 2
The base's radius is 2 cm.
7.G.B.4Work BackwardsAnswer
2 cm
Check the turns: a base of radius 2 cm has a circumference of 12.56 cm, and 12.56 × 5 = 62.8 cm, the whole path.
Takeaway
Rolling matches one circumference to another -- write that match down and it is the equation.
- Find the circle the cone traces
- Say what five turns cover
- Recover the radius