A cone can be created by cutting out a sector of a circle and taping the two radii together.[br]In the diagram of the cone, we can see the Pythagorean relationship between the height [math]h[/math], radius of the base of the cone [math]r[/math], and slant height of the cone [math]l[/math]:[br][br][math]r^2+h^2=l^2[/math]
If we fix the slant height [math]l[/math] (which corresponds to the radius of the flat sector, what is the radius [math]r[/math], in terms of [math]l[/math]?
[math]r=\frac{\alpha}{360^{\circ}}\cdot l[/math][br][br]Why is this true? We can use the arc length of the sector, which will become the circumference of the base:[br][math]\frac{\alpha}{360^{\circ}}\cdot2\pi l=2\pi r[/math][br]But then we can divide both sides by [math]2\pi[/math] to yield:[br][br][math]r=\frac{\alpha}{360^{\circ}}\cdot l[/math][br][br]
How does the central angle of the sector [math]\alpha[/math] relate to the angle between the altitude and slant height [math]\beta[/math]?
[math]\sin\beta=\frac{r}{l}=\frac{\frac{\alpha}{360^{\circ}}\cdot l}{l}=\frac{\alpha}{360^{\circ}}[/math][br][br][math]\sin\beta=\frac{\alpha}{360^{\circ}}[/math]