Question

A cone has a height of `5 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

`25pi+25sqrt2-(16pi+16sqrt2)=9pi+9sqrt2~=41 cm^2`

Explanation:

The surface area of a cone is:

`SA=pir(r+sqrt(h^2+r^2))`

We can calculate it for the cone at 5 cm height and at 4 cm height, subtract them, and thereby find the sliced off bit.

At h=5, r=5:

`SA=pi(5)(5+sqrt(5^2+5^2))=5pi(5+sqrt50)=25pi+5sqrt50=25pi+5sqrt25sqrt2=25pi+25sqrt2`

The angle of the cone doesn't change as we slice up the cone (it's 45 degrees), so the radius changes with the height. Therefore at h=4, r=4:

`SA=pi(4)(4+sqrt(4^2+4^2))=4pi(4+sqrt32)=16pi+4sqrt32=16pi+4sqrt16sqrt2=16pi+16sqrt2`

The difference is:

`25pi+25sqrt2-(16pi+16sqrt2)=9pi+9sqrt2~=41 cm^2`