Question

A cone has a height of `12 cm` and its base has a radius of `6 cm`. If the cone is horizontally cut into two segments `9 cm` from the base, what would the surface area of the bottom segment be?

Answer 1

`357.253\ \text{cm}^2`

Explanation:

Radius `r` of new circular section of bottom segment cut horizontally, at a height `h=9\ cm` from base, from an original cone of height `H=12\ cm` & base radius `R=6\ cm` is given by using property of similar triangles as follows

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

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

`=6(1-9/12)`

`=1.5\ cm`

Now, surface area of bottom segment of original cone

`=\text{area of circular top of radius 1.5 cm}+\text{curved surface area of frustum of cone}+\text{area of circular base of radius 6 cm}`

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

`=\pi(1.5)^2+\pi(1.5+6)\sqrt{9^2+(6-1.5)^2}+\pi(6)^2`

`=357.253\ \text{cm}^2`