Question

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?

Answer 1

`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`