2001 AMC 12 Problems/Problem 8 | Why is the Slant Height 10? The problem says:

Which of the cones listed below can be formed from a $252^\circ$ sector of a circle of radius $10$ by aligning the two straight sides? 

(A) A cone with slant height of 10 and radius 6
(B) A cone with height of 10 and radius 6
(C) A cone with slant height of 10 and radius 7
(D) A cone with height of 10 and radius 7
(E) A cone with slant height of 10 and radius 8

I looked at this question, I'm not sure what I should be visualizing. If I connect the two blue lines, that should form a circle with radius 7. So, couldn't I make the height of the cone any value? What I'm visualizing is squashing the circle into a smaller one, then having a line coming out of the paper as a height. Why does the radius of the original circle become the height of the cone? I’m very confused.
 A: This is the situation I faced when educating my son on this some years back. The solution - don't try to simply visualise it if you are having difficulty. Actually construct the shape and try folding it into a cone! And then you'll know exactly what's happening.
Any flat circular sector (which is basically like a circular pizza with a slice cut out of it) can be folded neatly into a three-dimensional right circular cone (without the circular base).
You will find that:


*

*it doesn't matter whether it's a minor sector (less than a semicircle) or a major sector (more than a semicircle) or even if it's exactly a semicircle. A cone can always be constructed. The centre of the sector will become the apex (peak or tip) of the cone. The smaller the sector, the more "peaky" the resulting cone. The larger the sector, the more "broad-based" the cone.

*the two radial edges of the sector become the slant height of the cone. The slant height is the length of the slanted edge measured from the base of the cone to its tip. It is not to be confused with the height of the cone, which is the perpendicular distance dropped from the tip of the cone directly through the centre of its base.

*the arc length of the sector becomes the circumference of the (open) circular base of the cone. This is the part a lot of students may find confusing. The sector, which is an incomplete circle folds to give a smaller complete circle (base of cone).

*the area of the sector is equal to the curved surface area of the cone.
So, in this case, the slant height is $10$ (equal to the radius of the sector) and the base radius can be computed using the arc length of the sector. 
If the radius of the sector is $R$ and the central angle is $\theta$ in degrees, its arc length is $\frac{\theta}{360}(2\pi R)$. 
If the base radius of the cone after folding is $r$, then you can equate the above expression to $2\pi r$.
Can you now complete the problem?
