# Find slant height of a cone

#### gazparkin

##### New member
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
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
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 $$\pi h^2$$ and circumference $$2\pi h$$. But the base of the cone had radius r so circumference $$2\pi r$$. The cone is only $$\frac{2\pi r}{2\pi h}= \frac{r}{h}$$ of the entire circle so has area $$\frac{r}{h}\pi h^2= \pi rh$$.