A cone has a height of #5 cm# and its base has a radius of #4 cm#. If the cone is horizontally cut into two segments #2 cm# from the base, what would the surface area of the bottom segment be?

1 Answer
Harish Chandra Rajpoot
Jul 11, 2018

#119.858\ \text{cm}^2#

Explanation:

Radius #r# of new circular section of bottom segment cut horizontally, at a height #h=2\ cm# from base, from a original cone of height #H=5\ cm# & base radius #R=4\ cm#, is given by using property of similar triangles as follows

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

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

#=4(1-2/5)#

#=2.4\ cm#

Now, surface area of bottom segment of original cone

#=\text{area of circular top of radius 2.4 cm}+\text{curved surface area of frustum of cone}+\text{area of circular base of radius 4 cm}#

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

#=\pi(2.4)^2+\pi(2.4+4)\sqrt{2^2+(4-2.4)^2}+\pi(4)^2#

#=119.858\ \text{cm}^2#

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