Question 476497
Surface Area of Cone: {{{pi*r^2 + pi*r*l}}}
where r is radius and l is slant height.
To find slant height, imagine a right triangle formed by points A,B,C where A is top of cone, B is point on outside of base and C is center of base.
The hypotenuse of this triangle is the slant height.
The legs are the vertical height and the radius.
Use Pythagorean theorem {{{a^2 + b^2 = c^2}}}
{{{5^2 + 19^2 = l^2}}}
{{{25 + 361 = l^2}}}
{{{386 = l^2}}}
Take square root of both sides
{{{19.647 = l}}}
Now substitute into surface area equation:
SA = {{{pi*5^2 + pi*5*19.647}}}
SA = {{{25pi + 98.23pi}}}
SA = {{{123.23pi}}}
SA = {{{387.15}}} 

------------------------------------------
For Lateral surface area, just take away the area of the base.
Area of base of cone is a circle: {{{pi*r^2}}}
r = 5
{{{pi*5^2 = 25pi}}}
Lateral SA = {{{387.15 - 25pi}}}
Lateral SA = {{{308.61}}}