A cone has a height of #15 cm# and its base has a radius of #8 cm#. If the cone is horizontally cut into two segments #8 cm# from the base, what would the surface area of the bottom segment be?

1 Answer

#579.047\ \text{cm}^2#

Explanation:

Radius #r# of new circular section of bottom segment cut horizontally, at a height #h=8\ cm# from base, from a original cone of height #H=15\ cm# & base radius #R=8\ cm# by using property of similar triangle we get

#\frac{R-r}{h}=\frac{R}{H}#

#r=R(1-\frac{h}{H})#

#=8(1-8/15)#

#=3.733\ cm#

Now, surface area of bottom segment of original cone

#=\text{area of circular top of radius 3.733}+\text{curved surface area of frustum of cone}+\text{area of circular base of radius 8}#

#=\pir^2+\pi(r+R)\sqrt{h^2+(R-r)^2}+\piR^2#

#=\pi(3.733)^2+\pi(3.733+8)\sqrt{8^2+(8-3.733)^2}+\pi(8)^2#

#=579.047\ \text{cm}^2#