Question

A cone has a height of `12 cm` and its base has a radius of `4 cm`. If the cone is horizontally cut into two segments `3 cm` from the base, what would the surface area of the bottom segment be?

Answer 1

The total surface area of the bottom segment is `148.081 cm^2`

Explanation:

Calculate the base area
`Pi*4^2 = 50.265`

Calculate the original sloping surface area
The formula for the sloping surface of a right cone is
`Pi*r*l` where `l` is the hypotenuse of the right angle formed by the base radius and the height.

For the current cone `l` can be calculated using pythagorus
`sqrt(12^2 + 4^2) = 12.649`
The original surface area is `Pi*12.649*4 = 158.953`

Calculate the new base radius
the height and the radius of the base give a right angled triangle with angle `theta` at the top of the cone.

equation 1: `tan theta = 4/12` (opposite over adjacent for a right angled triangle)

After the cut, `theta` and therefore `tan theta` doesn't change

The new height is 9cm (`12 - 3`)
let the new base radius be `x`

equation 2 :`tan theta = x/9`

Using equation 1 and equation 2 to eliminate `tan theta`
`x/9 = 4/12`

The new top radius is 3cm

Calculate the top surface of the bottom section
Using `Pi* r^2` to find the area

`Pi*3^2 = 28.274`

calculate the sloping surface of the new cone
`l = sqrt(9^2+3^2) = 9.487`
The new cone has a sloping surface area of `Pi*3*9.487 = 89.411`

Calculating the sloping surface area of the bottom section
Therefore the bottom section has a sloping surface area of `158.953 - 89.411 = 69.542` (original - top)

The total surface area is original base area + top area + sloping surface
`50.265 +28.274 +69.542 = 148.081`