A cone has a height of 12 cm and its base has a radius of 8 cm. If the cone is horizontally cut into two segments 4 cm from the base, what would the surface area of the bottom segment be?

2 Answers
Mar 8, 2018

S.A.=196pi cm^2

Explanation:

Apply the formula for the surface area (S.A.) of a cylinder with height h and base radius r. The question has stated that r=8 cm explicitly, whereas we would let h be 4 cm since the question is asking for S.A. of the bottom cylinder.
S.A.=2pi*r^2+2pi*r*h=2pi*r*(r+h)

Plug in the numbers and we get:
2pi*(8^2+8*4)=196pi

Which is approximately 615.8 cm^2.

You might think about this formula by imaging the products of an exploded (or unrolled) cylinder.
Unrolled cylinder diagram- CK-12 FoundationUnrolled cylinder diagram- CK-12 Foundation

The cylinder would include three surfaces: a pair of identical circles of radii of r that act as caps, and a rectangular wall of height h and length 2pi*r. (Why? Since when forming the cylinder the very rectangle would roll into a tube, precisely matching the outer rim of both circles that have circumferences pi*d=2pi*r.)

Now we find the area formula for each of the component: A_"circle"=pi*r^2 for each of the circle, and A_"rectangle"=h*l=h*(2pi*r)=2pi*r*h for the rectangle.

Adding them to find an expression for the surface area of the cylinder:
S.A.=2*A_"circle"+A_"rectangle"=2pi*r^2+2pi*r*h
Factor out 2pi*r to get S.A.=2pi*r*(r+h)

Notice that since each cylinder has two caps, there are two A_"circle" *in the expression for * S.A.

Reference and Image Attributions:
Niemann, Bonnie, and Jen Kershaw. “Surface Area of Cylinders.” CK-12 Foundation, CK-12 Foundation, 8 Sept. 2016, www.ck12.org/geometry/surface-area-of-cylinders/lesson/Surface-Area-of-Cylinders-MSM7/?referrer=concept_details.

Apr 15, 2018

:.color(purple)(=491.796cm^2 to the nearest 3 decimal places cm^2

Explanation:

:.Pythagoras: c^2=12^2+8^2

:.c=L=sqrt(12^2+8^2)

:. c=Lcolor(purple)(=14.422cm

:.12/8=tan theta=1.5=56^@18’35.7”

:.color(purple)(S.A.= pirL#

:.S.A.=pi*8*14.422

:.S.A.=362.464

:.Total S.A.color(purple)(=362.464cm^2

:.Cot 56^@18’35.7”*8=5.333cm=radius of top part

:.Pythagoras: c^2=8^2+5.333^2

:.c=L=sqrt(8^2+5.333^2)

:. c=Lcolor(purple)(=9.615cm top part
:.S.A. top part=pi*r*L

S.A. top part:.pi*5.333*9.615

S.A. top part:.=161.091

S.A. top part:.color(purple)(=161.091cm^2

:.S.A. Bottom partcolor(purple)(=362.464-161.091=201.373cm^2

:.S.A. Bottom part=201.373+89.361+201.062=491.796 cm^2
:.color(purple)(=491.796cm^2 to the nearest 3 decimal places cm^2