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Takeout Angle — Definition, Formula & Examples

Takeout Angle

The angle cut out of a piece of paper so that the paper can be rolled into a right circular cone.

 

Three shapes: a flat disk with a wedge cut out labeled "takeout angle," a cone being formed, and a complete right circular cone.

 

 

See also

Cone angle

Key Formula

δ=2π(1−rs)\delta = 2\pi\left(1 - \frac{r}{s}\right)
Where:
  • δ\delta = Takeout angle (in radians)
  • rr = Radius of the base of the cone
  • ss = Slant height of the cone (equal to the radius of the original flat disk)

Worked Example

Problem: A flat circular disk has radius 10 cm. You want to cut out a wedge and roll the remaining piece into a cone whose base has radius 6 cm. Find the takeout angle.
Step 1: Identify the given values. The slant height of the cone equals the radius of the original disk, so s = 10 cm. The base radius of the cone is r = 6 cm.
s=10,r=6s = 10,\quad r = 6
Step 2: Apply the takeout angle formula.
δ=2π(1−rs)=2π(1−610)\delta = 2\pi\left(1 - \frac{r}{s}\right) = 2\pi\left(1 - \frac{6}{10}\right)
Step 3: Simplify the fraction inside the parentheses.
δ=2π(1−0.6)=2π(0.4)=0.8π\delta = 2\pi\left(1 - 0.6\right) = 2\pi(0.4) = 0.8\pi
Step 4: Convert to degrees if needed.
δ=0.8π rad=0.8×180°=144°\delta = 0.8\pi \text{ rad} = 0.8 \times 180° = 144°
Answer: The takeout angle is 144° (or 0.8π radians). You remove a 144° wedge from the disk, and the remaining 216° sector rolls into the cone.

Another Example

Problem: You need a cone with base radius 5 cm and slant height 5 cm. What takeout angle should you cut?
Step 1: Since r = s = 5, the ratio r/s = 1.
rs=55=1\frac{r}{s} = \frac{5}{5} = 1
Step 2: Apply the formula.
δ=2π(1−1)=0\delta = 2\pi(1 - 1) = 0
Step 3: A takeout angle of 0 means no paper is removed, so the entire disk would be used. But r = s implies slant height equals base radius, which means the cone is completely flat — it's just a disk, not a cone at all. This confirms the formula: when r approaches s, the cone flattens out and no paper needs to be removed.
Answer: The takeout angle is 0°, meaning no wedge is removed and the shape stays flat — the limiting case of a completely flattened cone.

Frequently Asked Questions

How is the takeout angle related to the cone angle (apex half-angle)?
The cone's apex half-angle α satisfies sin α = r/s. The takeout angle is δ = 2π(1 − sin α). So a wider cone (larger α) has a smaller takeout angle, and a narrower, pointier cone (smaller α) has a larger takeout angle approaching 2π.
Why does the formula involve the ratio r/s?
When you roll a flat sector into a cone, the arc length of the sector becomes the circumference of the cone's base (2πr), while the sector's radius becomes the slant height s. The full circle has circumference 2πs, so the fraction of the circle you keep is r/s, and the fraction you remove is 1 − r/s. Multiplying by 2π gives the removed angle.

Takeout Angle vs. Cone Angle (Apex Angle)

The takeout angle δ is a property of the flat pattern (net) of a cone — it is the angle of the wedge you remove from a disk. The cone angle (or apex angle 2α) is a property of the three-dimensional cone itself, measuring the full opening at the tip. They are related by δ = 2π(1 − sin α), but they describe different geometric objects: one is flat, the other is spatial.

Why It Matters

The takeout angle is essential in sheet-metal fabrication, packaging design, and any manufacturing process where a flat material must be cut and formed into a conical shape. Knowing this angle lets you compute the exact pattern to cut from a flat sheet, minimizing waste. It also appears in geometry and calculus courses when studying the relationship between a cone and its lateral surface.

Common Mistakes

Mistake: Confusing the takeout angle with the angle of the remaining sector.
Correction: The takeout angle is the angle of the piece you remove. The remaining sector has central angle 2π − δ = 2π(r/s). Make sure you know which angle the problem asks for.
Mistake: Using the cone's height h instead of its slant height s in the formula.
Correction: The flat disk's radius equals the cone's slant height, not the vertical height. If you know h and r, first compute s = √(r² + h²), then use that in the formula.

Related Terms