Question

find the surface area and the volume of the following cone.
(1) surface area:
(2) volume:

Explanation:

Step1: Calculate the slant height $l$

Using the Pythagorean theorem $l=\sqrt{r^{2}+h^{2}}$, with $r = 5$ and $h=12$. So $l=\sqrt{5^{2}+12^{2}}=\sqrt{25 + 144}=\sqrt{169}=13$.

Step2: Calculate the surface area of the cone

The surface - area formula of a cone is $A=\pi r(r + l)$. Substitute $r = 5$ and $l=13$ into the formula: $A=\pi\times5\times(5 + 13)=5\pi\times18 = 90\pi$.

Step3: Calculate the volume of the cone

The volume formula of a cone is $V=\frac{1}{3}\pi r^{2}h$. Substitute $r = 5$ and $h = 12$ into the formula: $V=\frac{1}{3}\pi\times5^{2}\times12=\frac{1}{3}\pi\times25\times12 = 100\pi$.

Answer:

(1) $90$
(2) $100$