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Net of a Cone — Definition, Formula & Examples

A net of a cone is the flat, 2D shape you get when you 'unfold' a cone and lay it flat. It consists of two pieces: a circular base and a sector (wedge shape) that forms the lateral surface.

The net of a right circular cone is a planar figure composed of a circle of radius rr (the base) and a circular sector of radius equal to the slant height ll, whose arc length equals the circumference of the base, 2πr2\pi r.

Key Formula

θ=rl×360°\theta = \frac{r}{l} \times 360°
Where:
  • θ\theta = Central angle of the sector in degrees
  • rr = Radius of the cone's circular base
  • ll = Slant height of the cone (radius of the sector)

How It Works

To build a net of a cone, start by drawing a circle for the base. Then draw a sector (a 'pizza slice' shape) whose radius equals the slant height of the cone. The curved edge of the sector must match the circumference of the base circle, so the central angle of the sector is θ=rl×360°\theta = \frac{r}{l} \times 360°, where rr is the base radius and ll is the slant height. When you cut out both pieces and fold the sector into a cone shape, its straight edges meet and its curved edge lines up perfectly with the circular base.

Worked Example

Problem: A cone has a base radius of 3 cm and a slant height of 9 cm. Find the central angle of the sector in its net.
Identify the values: The base radius is r=3r = 3 cm and the slant height is l=9l = 9 cm.
Apply the formula: Substitute into the sector angle formula.
θ=39×360°=13×360°=120°\theta = \frac{3}{9} \times 360° = \frac{1}{3} \times 360° = 120°
Describe the net: The net consists of a circle with radius 3 cm and a sector with radius 9 cm and a central angle of 120°.
Answer: The sector in the net has a central angle of 120°.

Why It Matters

Understanding cone nets is essential for calculating lateral surface area, since the area of the sector gives πrl\pi r l directly. It also appears in design and manufacturing whenever conical shapes—like funnels, party hats, or ice cream cones—need to be cut from flat material.

Common Mistakes

Mistake: Confusing the slant height with the vertical height when drawing the sector.
Correction: The radius of the sector is the slant height ll, not the altitude hh of the cone. If only the altitude and base radius are given, find the slant height using l=r2+h2l = \sqrt{r^2 + h^2}.