Problem
Central angle of a cone net
The net of a cone is drawn. The sector becomes the side and the circle becomes the base. The sector's radius is 9 cm and the base circle's radius is 3 cm. Find the sector's central angle.
Reasoning · Grade 6-2 Shortest Distance and Surface Area
Your answer
Watch out Comparing the radii is tempting, but what must match is the arc and the rim.
How to solve
Strategy Create a Physical Representation — Cut the two pieces out and try to build the cone: the only way the sector's curved edge can be taped all round the rim of the base circle is if those two curves have the same length. That single sentence turns an angle question into a length question, and lengths are something you can compute. Then, rather than attack the sector head-on, solve the easier related problem first — how long is the rim of a whole circle of radius 9 cm? — and ask what fraction of it the sector's arc is. That fraction is the same fraction of 360 degrees.
1STEP 1
See what the folding forces
Folded, the arc meets the base's rim.
(arc of the sector) = (circumference of the base circle)
This is the one idea in the problem, and paper makes it obvious: if the curved edge were too long the cone would buckle, and if it were too short there would be a gap.
6.G.A.4Create A Physical RepresentationFolding forces the sector's curved edge to be exactly as long as the base circle's rim.
Why?
Rolling the paper does not stretch it, so the curved edge keeps its length as it wraps around the base.
🧱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 rim is met with no gap and no overlap, so the two lengths must be equal or the cone would buckle or leave a slit.
🧱Whole is the sum of its partsBreak a thing into pieces with no gaps or overlaps and the pieces add back to the whole.
2STEP 2
Measure the base circle's rim
The base's rim is 18.84 cm.
3 × 2 × 3.14 = 18.84 cm
Circumference is a length in centimetres, the same kind of quantity as the arc it has to match, so it makes sense to set the two equal.
7.G.B.4Analyze The Units3STEP 3
Solve the easier related problem: a whole circle of radius 9 cm
A full circle of radius 9 is 56.52 cm.
9 × 2 × 3.14 = 56.52 cm
Working out the whole first and then asking 'what fraction of it do I need?' is easier than trying to build the piece directly.
7.G.B.4Solve An Easier Related Problem4STEP 4
Find what fraction of the whole circle the sector is
The sector is a third of the whole circle.
18.84/56.52 = 1/3 → (angle)/360° = 1/3
Arc length and central angle are proportional — doubling the angle doubles the arc — so one ratio can simply be swapped for the other.
7.RP.A.2Look For A Pattern5STEP 5
Take a third of a full turn
A third of a full turn is 120 degrees.
(angle) = 360° × 1/3 = 120°
A whole turn is 360 degrees, so a third of a turn is 360 divided by 3 — no protractor needed.
4.MD.C.5Draw A Diagram6STEP 6
Notice that 3.14 was never actually needed
The ratio of radii alone also gives 120 degrees.
(angle) = 360° × (base radius)/(slant height) = 360° × 3/9 = 120°
Both lengths are built the same way out of pi, so pi is a common factor that cancels — a good reminder to look for cancelling before reaching for a calculator.
7.RP.A.2Analyze The UnitsAnswer
120 degrees
360 × 3 ÷ 9 = 120
120 degrees is between 0 and 360, so it can genuinely be a sector's central angle, and it is a third of a full turn — which matches the picture, where the sector is clearly a good deal more than a quarter circle but well short of a half circle. Check the lengths directly: a 120-degree sector of radius 9 cm has an arc of 9 x 2 x 3.14 x 120/360 = 56.52/3 = 18.84 cm, and the base circle's circumference is 3 x 2 x 3.14 = 18.84 cm. They match, so the net really does close up. The shortcut agrees too: 360 x 3/9 = 120. And it passes a sanity test on size — the base radius is a third of the slant height, so the sector must be a third of a circle; if the base were bigger the angle would be bigger, and a base radius equal to the slant height would need the whole 360 degrees, meaning a flat disc rather than a cone.
Takeaway
The curved edge of the flat piece has to wrap exactly round the base circle, so the angle is just 360° times base radius over slant height — and pi cancels right out.
- See what the folding forces
- Measure the base circle's rim
- Solve the easier related problem: a whole circle of radius 9 cm
- Find what fraction of the whole circle the sector is
- Take a third of a full turn
- Notice that 3.14 was never actually needed