# 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?

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Mar 8, 2018

#### Answer:

$S . A . = 196 \pi$ $c {m}^{2}$

#### 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 $r = 8$ $c m$ explicitly, whereas we would let $h$ be $4$ $c m$ since the question is asking for $S . A .$ of the bottom cylinder.
$S . A . = 2 \pi \cdot {r}^{2} + 2 \pi \cdot r \cdot h = 2 \pi \cdot r \cdot \left(r + h\right)$

Plug in the numbers and we get:
$2 \pi \cdot \left({8}^{2} + 8 \cdot 4\right) = 196 \pi$

Which is approximately $615.8$ $c {m}^{2}$.

You might think about this formula by imaging the products of an exploded (or unrolled) cylinder.

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 $2 \pi \cdot r$. (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 $\pi \cdot d = 2 \pi \cdot r$.)

Now we find the area formula for each of the component: ${A}_{\text{circle}} = \pi \cdot {r}^{2}$ for each of the circle, and ${A}_{\text{rectangle}} = h \cdot l = h \cdot \left(2 \pi \cdot r\right) = 2 \pi \cdot r \cdot h$ for the rectangle.

Adding them to find an expression for the surface area of the cylinder:
$S . A . = 2 \cdot {A}_{\text{circle"+A_"rectangle}} = 2 \pi \cdot {r}^{2} + 2 \pi \cdot r \cdot h$
Factor out $2 \pi \cdot r$ to get $S . A . = 2 \pi \cdot r \cdot \left(r + h\right)$

Notice that since each cylinder has two caps, there are two ${A}_{\text{circle}}$ *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.

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