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?

1 Answer
Feb 15, 2017

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#