A cone has a height of #8 cm# and its base has a radius of #6 cm#. If the cone is horizontally cut into two segments #7 cm# from the base, what would the surface area of the bottom segment be?

1 Answer
May 24, 2017

Area is #765/8pi# #cm^2~~300.42# #cm^2#

Explanation:

http://blog.easycareinc.com/blog/if-the-shoe-fits/the-hoof-is-frusto-what Figure 1

So we have a cone that has been sliced into 2. The bottom cone consists of two circles and a frustum. The flattened frustum can be seen on the right side of the image below.

The Gizmologist's Lair Figure 2

We can see that the area of the frustum is just the difference between the area difference of the sector 2 concentric circles.
This area difference is essentially the formula for the area of a frustum:
#A=pi(R+r)sqrt((R-r)^2+h^2)#

Ignoring Figure 2 and its labels, #R# here is the big circle's radius and #r# is the smaller circle's radius, and #h# is the height of the frustum.

Back to Figure 1, the height #h# of the bottom portion of the cone is #7cm# as stated in the problem, and the radius #R# of the bottom is #6cm#. Now we just need to find #r#.

Recall #tantheta=(opposite)/(adjacent)#

In the cone, image #2theta# as the angle at the vertex of the cone (refer to Figure 1). In that case, the opposite side would be the radius of the cone, #R# and the adjacent side would be the height of the cone, #h#.

#tantheta=R/h=6/8=3/4#

However, the opposite side of #theta# can also be #r#, the radius of the top of the frustum and the adjacent side can also be #h-7cm# or #1#. In this case:

#tantheta=r/(h-7)=r#

Substitute #tantheta=3/4#:
#r=3/4#

Phew!

Now, we can finally calculate the area of the frustum! #A=pi(R+r)sqrt((R-r)^2+h^2)#
where #R=6cm#, #h=7cm#, #r=3/4cm#
#A=(945)/(16)pi# #cm^2#

Now we can calculate the areas of the top and bottom circles easily and add it all together:

#A_("total")=945/16pi+9/16pi+36pi#
#=pi(945/16+9/16+36)#
#=765/8pi# #cm^2#
#~~300.42# #cm^2#

#:.# The surface area of the bottom portion is around #300.42# #cm^2#