A cone has a height of #6 cm# and its base has a radius of #6 cm#. If the cone is horizontally cut into two segments #3 cm# from the base, what would the surface area of the bottom segment be?

Question

A cone has a height of #6  cm# and its base has a radius of #6  cm#.  If the cone is horizontally cut into two segments #3 cm# from the base, what would the surface area of the bottom segment be?

Claire Akin · Accepted Answer

Yosief you answer is great but you just subtracted the original cone and the top cone. While that will give you a complete solution for the volume for the surface area you have three parts you need to consider:
1) The Base Area => this area of the circle
2) The side Area, that is the one you got by #pis(R-r)# that you have right
3) Because you cut it now you have a smaller circle on top so you need to this area.

So using your terms
#SA_F = "BA + Side Area + Top Area"#
#SA_F = piR^2 + pis(R-r) + pir^2 = pi(R^2+r^2) + pis(R-r)#

What changed? Well in your formula the first is the sum of squares instead the difference of squares. You were not that far you were a sign away from the correct answer. Make that change and voila you got it...

Savannah Aguilar · Answer

Surface area of bottom shape = #9pi(4 + sqrt18)#

Let's start off by writing down what is already known and then finding some of the values we will need later.

The BIG cone and the small cone are similar figures. R = 6cm ; r = 3cm H = 6cm ; h = 3cm L = #sqrt72# ; # l# = #sqrt18# (#l# is the slant height, found by Pythagoras)

Small cone: surface area = area base + curved surface s.a. = #pi r^2# + #pi r l# = #pi(3)^2 + pi (3)sqrt18#

Factorise to make it easier: s.a. = #3pi(3 + sqrt18)#

In similar figures, the areas are in the same ratio as the square of the sides, length, radius etc.

#3^2/6^2# = #(3pi(3 + sqrt18))/(SA)# #rArr# #1/4# = #(3pi(3 + sqrt18))/(SA)#

BIG CONE: Surface area = #4 xx 3pi(3 + sqrt18)# = #12pi(3 + sqrt18)#

Surface area of fustrum (bottom segment);

BIG AREA - small area

= #12pi(3 + sqrt18)# - #3pi(3 + sqrt18)#

= #9pi(3 + sqrt18)#

However, the circular cut surface must be included as well.

Total Surface area = #9pi(3 + sqrt18)# + #pi(3)^2# = #9pi(3 + sqrt18)# + #9pi# = #9pi(3 + sqrt18 +1)# = #9pi(4 + sqrt18)#

Sadie Allen · Answer

undefined To find the surface area of the bottom segment of the cone, you first need to calculate the slant height of the cone using the Pythagorean theorem. The slant height (l) can be found using the formula:

l = √(r^2 + h^2)

where r is the radius of the base and h is the height of the cone.

Given:
Radius (r) = 6 cm
Height (h) = 6 cm

Using the formula:

l = √(6^2 + 6^2) 
  = √(36 + 36) 
  = √72 
  ≈ 8.49 cm

Now, since the cone is cut horizontally 3 cm from the base, the radius of the smaller cone (formed by the cut) is also 6 cm, and its height is 3 cm.

Using the same formula for the slant height (l) for the smaller cone:

l' = √(6^2 + 3^2)
   = √(36 + 9)
   = √45
   ≈ 6.71 cm

The surface area of the bottom segment of the cone can be calculated using the formula:

Surface Area = πr'l' + πr^2

where r' is the radius of the smaller cone and l' is its slant height.

Surface Area = π(6)(6.71) + π(6^2)
             ≈ 126.98 cm^2 + 113.10 cm^2
             ≈ 240.08 cm^2

Therefore, the surface area of the bottom segment of the cone is approximately 240.08 square centimeters.