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Find slant height of a cone

gazparkin

New member
Jul 1, 2019
17
Hello,

Could anyone help me understand the steps on the below questions?

A cone has a total surface area of 300π cm² and a radius of 10 cm. What is its slant height?


A cone has a slant height of 20 cm and a curved surface area of 330 cm2. What is the circumference of its base?


I'd really like to know what steps I need to take to get to the answer on these.

Thank you in advance :)
 

skeeter

Well-known member
MHB Math Helper
Mar 1, 2012
643
Hello,

Could anyone help me understand the steps on the below questions?

A cone has a total surface area of 300π cm² and a radius of 10 cm. What is its slant height?


A cone has a slant height of 20 cm and a curved surface area of 330 cm2. What is the circumference of its base?


I'd really like to know what steps I need to take to get to the answer on these.
(1) total surface area = lateral surface area + base area

$A_T = \pi r s + \pi r^2$, where $s$ is the slant height and $r$ is the base radius

solving for $s$ $\implies s = \dfrac{A_T - \pi r^2}{\pi r}$

(2) assuming "curved surface area" is the lateral surface area ...

$A_L = \pi r s \implies r = \dfrac{A_L}{\pi s}$

use the formula for a circle's circumference to finish
 

HallsofIvy

Well-known member
MHB Math Helper
Jan 29, 2012
1,151
Suppose a cone (minus the circular bottom) has radius r and slant height s. Cut a slit along the slant and flatten it (Unlike a sphere a cone can be flattened. it is a "developable surface."). It will form part of a circle with radius h. That entire circle has radius h so area [tex]\pi h^2[/tex] and circumference [tex]2\pi h[/tex]. But the base of the cone had radius r so circumference [tex]2\pi r[/tex]. The cone is only [tex]\frac{2\pi r}{2\pi h}= \frac{r}{h}[/tex] of the entire circle so has area [tex]\frac{r}{h}\pi h^2= \pi rh[/tex].