Question

A cone has a height of `8 cm` and its base has a radius of `5 cm`. If the cone is horizontally cut into two segments `1 cm` from the base, what would the surface area of the bottom segment be?

Answer 1

`173.403\ \text{cm}^2`

Explanation:

Radius `r` of new circular section of bottom segment cut horizontally, at a height `h=1\ cm` from base, from an original cone of height `H=8\ cm` & base radius `R=5\ cm` is given by using property of similar triangles as follows

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

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

`=5(1-1/8)`

`=4.375\ cm`

Now, surface area of bottom segment of original cone

`=\text{area of circular top of radius 4.375 cm}+\text{curved surface area of frustum of cone}+\text{area of circular base of radius 5 cm}`

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

`=\pi(4.375)^2+\pi(4.375+5)\sqrt{1^2+(5-4.375)^2}+\pi(5)^2`

`=173.403\ \text{cm}^2`