A sector of 120 degrees is cut off from a circle and the two ends are joined to form a cone.Find the semi vert

Question

A sector of 120 degrees is cut off from a circle and the two ends are joined to form a cone.Find the semi vertical angle of the cone.

Answer 1

I am guessing you are talking about the angle you form by passing a plane vertical to the base through the the top vertex. Say the original vertex had radius R and you took out a sector of 120 degrees, then the curve at the end of the segment you took out has length 2(Pi)R/3 as 120 degrees is 1/3 of 360 degrees. Now when you make a cone, that curve becomes the perimeter of the base circle of the cone. So if r denotes the radius of that base circle we have
2(Pi)r=2(Pi)R/3
so after cancellations you have
r=R/3.
Now if you draw a line from the top vertex of the cone to the center of the circle you obtain a right triangle with one short side of length r=R/3 and hypotenuse R. So the angle at the top is
arcsin(R/3/R)=arcsin(1/3)
and as the picture is symmetric to get the angle you need you just need to multiply by two, so the answer is
2arcsin(1/3).

Answer 2

What's the semi-verical angle? I don't know how to get that, but you can get the height, radius, and slant height of the cone. Say that the radius of the original circle is r. When the 120 degree section is turned into a cone, the slant height of the cone is r. Say z is the length of the perimeter of the circle pertaining to the 120 degree section. That means z is (120/360) * 2 * pi * r, or (2/3) * pi * r. So the perimeter of the bottom of the cone is (2/3) * pi * r. That means that the radius of the cone is (1/3)r. With the slant height and the radius, you can calculate the height using the Pythagorean Theorem, which gives you (r ^ 2 - (1/9)r ^ 2) ^ .5, or ((8/9)r ^ 2) ^ .5, which is [ (8 ^ .5) / 3 ]r.

Answer 3

The information required for a cone

V = volume

S = total surface area

r = radius

h = altitude ( height )

s = slant height

Answer 4

some parameters are missing, such as, radius of circle,etc...