# A cone has a height of 14 cm and its base has a radius of 7 cm. If the cone is horizontally cut into two segments 2 cm from the base, what would the surface area of the bottom segment be?

Jan 20, 2017

$358 c {m}^{2}$

#### Explanation:

Solution: Subtract the top cut outer area from the original to derive the bottom section outer area. Calculate the new circular (cut) section area and the base area. Combine the three values to the final area of the bottom segment.

The surface area of the cone equals the area of the circle plus the area of the cone and the final formula is given by:

SA = $\left(\pi \cdot {r}^{2}\right) + \left(\pi \cdot r \cdot l\right)$

Where,
h is the height
l is the slant height

The area of the curved (lateral) surface of a cone = πrl
NEW AREA = Original Lateral Surface – Top Cut Lateral Surface + Original Base + New Cut top (top cut base)
Original slant height was ${l}^{2} = {r}^{2} + {h}^{2}$ ; l^2 = 7^2 + 14^2 = 24 l = 15.65

A horizontal cut means simply that the cone top now has a height of 12cm.
The original cone had an angle with a tangent of $\frac{14}{7.}$ Therefore, the new lengths must have an equal tangent of $\frac{12}{x} .$
So, ${r}_{2} = \left(\frac{12}{14}\right) \cdot 7 = 6 c m$

The area of the circle is $\pi \cdot {\left(6\right)}^{2} = 113$
Top Cut cone slant height is ${l}^{2} = {r}^{2} + {h}^{2}$ ; l^2 = 6^2 + 12^2 = 36 + 144 ; l = 13.4
Original Base area is $\pi \cdot {\left(7\right)}^{2} = 154$

NEW AREA = Original Lateral Surface – Top Cut Lateral Surface + Original Base + New Cut top (top cut base)
NEW AREA = $\left(\pi \cdot 7 \cdot 15.65\right) - \left(\pi \cdot 6 \cdot 13.4\right) + \left(\pi \cdot {\left(7\right)}^{2}\right) + \left(\pi \cdot {\left(6\right)}^{2}\right)$

NEW AREA = 344 – 253 + 154 + 113 = $358 c {m}^{2}$