Question

A cone has a height of `16 cm` and its base has a radius of `3 cm`. If the cone is horizontally cut into two segments `7 cm` from the base, what would the surface area of the bottom segment be?

Answer 1

By cutting off a segment of a cone parallel to the base, you create what is known as a frustum. There are four important pieces of information in a fustrum: The height `(h)`, the larger radius `(R_1)`, the smaller radius `(R_2)` and the slant `(s)`. In the question asked we know two of these variables, h=7 and R1=3. The first thing we need to do is find `R_2`.

By looking at the question, we see the original cone has a height of 16cm and a radius of 3cm. This means the relationship between the height and the radius is equal to `16/3`. In the frustum, we have the height, but the smaller radius is unknown, which makes the relationship of height to radius equal to `7/R_2`. Since the ratios of the cone have been unchanged while making it a fustrum, we can safely say that the height-radius ratio of the cone is the same in the fustrum, so
`16/3=7/R_2`. By cross-multiplying, we find that
`21=16R_2`. Divide both sides by 16, and we get
`21/16=R_2`
`1.3125=R_2`

Now we have the values of `h, R_1 and R_2`, all that is left is `s`. The formula for finding `s` is as follows:
`s=sqrt((R_1-R_2)^2+h^2)`
By looking at the image below, you should get an understanding of how this works using Pythagoras' Theorem.

http://www.ditutor.com/solid_gometry/frustum_cone.html
Now we simply plug in the values we have into the formula, to get
`s=sqrt((3-1.3125)^2+7^2)`
`s=sqrt(2.8476+49)`
`s=sqrt(51.8476)`

Now we know all the values necessary to calculate the Surface Area, the formula for which is as follows:
`A_s=pi(s(R_1+R_2)+(R_1)^2+(R_2))`. This formula is derived from showing the 2-D map of a fustrum, such as below, where the bottom circle has `R_1`, and the top circle has `R_2`.

http://www.ditutor.com/solid_gometry/frustum_cone.html

Lastly, we plug in all our known values into the equation to get
`A_s=pi(sqrt(51.8476)(3-1.3125)+3^2+(1.3125)^2)`
`A_s=pi(12.151+9+1.7226)`
`A_s=pi(22.874)`
`A_s~~71.859cm^2`
Therefore the surface area of the bottom segment of the cone is roughly 71.859`cm^2`.

I hope I helped