Cube3D 立方体数理Maths · Middle school / High school
Why does the side of a cone unfold into a sector?
Imagine removing the base disk from a paper cone, then cutting along the seam from the rim to the tip. In the experiment, move the unfolding slider slowly and keep your eye on the orange rim. Once the side lies flat, the rim forms the curved edge of a sector, with the cone’s tip at its centre.
Match the parts before and after unfolding
Every line from the cone’s tip straight down its side to the rim has the same length. This slant height ℓ becomes the radius of the flat sector. The vertical height h goes straight to the centre of the base. It forms a right triangle with the base radius r and the slant height, so ℓ = √(r² + h²).
The sheet does not stretch. The orange rim starts at length 2πr and keeps that length as the sector arc. With r = 3 cm and h = 4 cm, ℓ = 5 cm and the rim is 6π cm, about 18.85 cm. A full circle of radius 5 cm has circumference 10π cm, so the sector takes up 3/5 of a full turn: 216°.
Predict the effect of making the cone taller
The sector occupies the fraction r/ℓ of a circle of radius ℓ. Its area is (r/ℓ) × πℓ² = πrℓ. For the starting cone, the area is 15π cm², about 47.12 cm². This is the area of the side; the base disk is a separate piece.
Flatten the cone and keep r at 3 cm. Increase h from 4 cm to 8 cm. The arc still measures 18.85 cm, but the angle shrinks from 216° to about 126.4°. The area grows to about 80.53 cm². The same length of arc takes up a smaller fraction of a bigger circle, while the distance from the tip to the entire rim increases.
Check your understanding
Double both starting dimensions to r = 6 cm and h = 8 cm. Predict the angle and compare the new side area to the original.
Show the explanation
The slant height doubles to 10 cm, so r/ℓ is still 3/5 and the sector angle stays 216°. Both factors in πrℓ double, giving 60π cm², four times the original 15π cm².
What the model assumes
The simulation represents the flexible side of an ideal right circular cone. It ignores paper thickness, overlap at the seam and material stiffness. The base disk is not part of the drawing. The view rescales to fit, so use the readouts for length and area comparisons. The interactive experiment requires Family Plus or Teacher access.