Question

find the surface area and the volume of the following cone.
(1) surface area: ____π sa = πr²+πrs
r =
s =
(2) volume: ____π v = ⅓πr²h

Explanation:

Step1: Find the radius $r$

The diameter of the base of the cone is 12, so $r=\frac{12}{2}=6$.

Step2: Find the slant - height $s$

We know the height $h = 5$ and $r = 6$. Using the Pythagorean theorem for a cone $s=\sqrt{r^{2}+h^{2}}=\sqrt{6^{2}+5^{2}}=\sqrt{36 + 25}=\sqrt{61}$.

Step3: Calculate the surface area

The surface - area formula of a cone is $SA=\pi r^{2}+\pi rs$. Substitute $r = 6$ and $s=\sqrt{61}$ into the formula:
$SA=\pi\times6^{2}+\pi\times6\times\sqrt{61}=36\pi+6\sqrt{61}\pi$.

Step4: Calculate the volume

The volume formula of a cone is $V=\frac{1}{3}\pi r^{2}h$. Substitute $r = 6$ and $h = 5$ into the formula:
$V=\frac{1}{3}\pi\times6^{2}\times5=\frac{1}{3}\pi\times36\times5 = 60\pi$.

Answer:

(1) $(36 + 6\sqrt{61})\pi$
(2) $60\pi$