MODELING:
Sharpen a six sections cylinder, the most important part - Standard execution time: 2 minute.

1
Create a Cylinder with 6 sections.
Select the top (end) face and similar bottom face.

RMB>LMB:Cylinder:[]:6
LMB:select face i

2
We add vertices to have a better control to the final body
Select edges, cut each in six parts and connect them.

e 6 c

3
Deselect all vertices.
In Face mode, select all faces and,
with 'Ctrl' key pressed, deselect all the vertical faces

space f CtrlA  Ctrl+LMB:deselect

4
Now 'Dissolve' to have just two faces again at top and bottom and Deselect all.

<— space

5
In Y view, select the top face and make a little (<5%) Inset to create a new face.
(Figure 5-7 show a bigger inset for clarity)

y LMB:select
RMB>LMB:Inset  Tab 1 Ret

6
Select all vertices of the face and begin an advanced 'Inflate-to-circle' operation
Here 'advanced' mode is not mandatory, but more accurate.

v RMB>LMB:Deform RMB:Inflate

7
To pick the center of 'Inflate', select the top/center face
- a blue dot mark the center - Right clic to valid.

LMB:select center RMB:clic

8
Now pick the radius point of 'Inflate' (choose outer vertex on x axis) and Right clic to valid
(easier with a MMB zoom if 1% inset)

LMB:select radius RMB:clic

9
Make a perfect (?!) circle... his diameter equal the cylinder diameter.

tab 100 Ret

10
Return in standard view, deselect all and select the new (top) circular face.

r space LMB:select

11
Scale the face to 50% (diameter of the lead)...

RMB>LMB:Scale Uniform
tab 50 Ret

12
Move the face along y axis to 3 wu.
Step 12 can be made before step 11, and vice versa

RMB>LMB:Move:y tab 3 Ret

13
Deselect all and, on the center edge loop, select a vertex (as shown) just in center of one cylinder side; then select similar vertices.

space v LMB:select vertex i

14
Now beginning an 'advanced' move similar to
the 'Inflate'-to-circle but with magnet and axis constraint:
Pick a vertical edge to choose Y axis as Move axis.
(or pick the top circular edge)

RMB>RMB:Move LMB:select axis

15
Now pick vertices for Magnet influence:
Alt-Right clic and select all vertices of the center loop.
A blue dot mark the center of the influence zone.

Alt+RMB:clic LMB:select vertices

16
Then choose a Magnet Option.
Here 'Shortest' or 'Surface'.

Alt+RMB:clic LMB:select 'Surface'

17
Right clic to valid options
and move vertices to -1.5 wu

RMB:clic Tab -1.5 Ret

Here value is negative because we choose the vertical edge under moving vertices in step 14; if you pick the top circular edge, the value is positive.

NB: They are many way to make this 'advanced' move, but step 15-17 are preferred for simplicity in key sequence and efficiency. Another way is to select just one outer vertex on the loop for Magnet influence, choose 'MidPoint' as Magnet option, then Right clic to valid options and move vertices with the mouse (D <1 wu). In this case the final look of the pencil is slightly different but all sharpener are different...

18
Deselect all. In edge mode Select all edges, then Deselect all vertical edges
and change the 'Hardness' to... hard.
Just to have a good preview, later we reverse this step...

space e CtrlA
Ctrl+LMB:deselect all vertical edges
RMB>LMB:Hardness:Hard

18bis
Preview this (not so) hard work.
(If needed, to adjust the cone aspect, you can move slightly on y axis the top end face). Then return in Workmode.

space Tab MMB:turn the view tab

Time to save, no? Now some 'Finishing' by the Menu...