Depth · one concept from first principles · step 2 of 4

Match the traced circle's circumference to the cone

A cone whose slant height is $10\ \text{cm}$ is rolled about its apex, as shown at the right. After $5$ turns it is back where it started. What is the radius of the cone's base, in $\text{cm}$ ?

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 touching point of the base stays one slant height from it, so it travels round a circle of radius 10 cm.

10 × 2 × 3.14 = 62.8

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

3STEP 3

Recover the radius

Divide the circumference back out by 2 and by 3.14.

12.56 ÷ 3.14 = 4, 4 ÷ 2 = 2

Answer

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

Where next?

Another one like thissuggested

12 practice problems

Same look, different logicsuggested

Find the matching lengths in a cylinder net

Step back and solidifysuggested

One full roll moves it its own circumference A part's share of a circle is its central angle's share

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▶ Practice — 12 problems