Answer to Question #197963 in Geometry for Angelo

Question

A circular cone piece of cardboard with a diameter of 1 m will be made into a conical hat 44 cm high by cutting a sector off and joining the edges to form a cone. Determine the anglein degree subtended by the sector removed.

Answer 1

$$r=100\ cm/2=50\ cm, h=44\ cm$$

From the right triangle

$$x=\sqrt{r^2-h^2}=\sqrt{50^2-44^2}=2\sqrt{141}(cm)$$

Let:

$C_1=$ circumference of the circle

$$C_1=2\pi r$$

$C_2=$ circumference of the base of the cone

$$C_2=2\pi x$$

$C=$ length of the arc

$$C=r\theta$$

Then

$$C=C_1-C_2$$

$$r\theta=2\pi r-2\pi x$$

$$\theta=\dfrac{2\pi(r-x)}{r}$$

$$\theta=2\pi(1-\dfrac{x}{r})$$

$$\theta\degree=2\pi(1-\dfrac{x}{r})(\dfrac{360\degree}{2\pi})$$

$$\theta\degree=(1-\dfrac{x}{r})360\degree$$

$$\theta\degree=(1-\dfrac{2\sqrt{141}}{50})360\degree\approx189\degree$$