A cone has a height of #36 cm# and its base has a radius of #9 cm#. If the cone is horizontally cut into two segments #24 cm# from the base, what would the surface area of the bottom segment be?

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Answer 1 · 2017-05-23T11:00:29

The surface area of the bottom segment #= 256.259" cm"^2#

To explain this, I will show it with a diagram of a triangle as if the cone was vertically split.

enter image source here

#"Surface area of a cone" = pi r^2 + pi r s#

#"SA" = pi "DG"^2 + pi r"DG" xx"DE"#

We can find out the length of line #DG# because since lines #CE# and #BE# are of constant gradient, #"CH"/"CE" = "DG"/"DE"#

#"CH"/"CE" = "DG"/"DE"#

#4.5/"CE" = "DG"/"DE"#

#"CE"^2 = 4.5^2 + 36^2# (Pythagoras theorem)

#"CE"^2 = 20.25 + 1296#

#"CE" = sqrt(1316.25)#

#"CE" = 36.28#

#4.5/"36.28" = "DG"/"DE"#

#"HE"/"CE" = "GE"/"DE"#

#"36"/"36.28" = "24"/"DE"#

#"36"/"36.28" = "24"/"24.19"#

now we know that:

#"DE" = 24.19#

So we can now use the pythagorean theorem again to find out what #"DG"# is.

#"DG"^2 = "DE"^2 - "GE"^2#

#"DG"^2 = 24.19^2 - 24^2#

#"DG" = sqrt(585 - 576#

#"DG" = sqrt(9#

#"DG" =3#

We can also do the same equation as above to find out what #"DG"# is, but this was what first came to my mind when answering this question

Now that we have those measurements, we can substitute them in the Surface area equation.

#"SA" = pi 3^2 + pi 3 xx 24.19#

#"SA" = 28.274 + 9.425 xx 24.19#

#"SA" = 28.274 + 227.985#

#"SA" = 256.259" cm"^2#