Here we simply need to remember the formula for the surface area of a cone and plug in our values for the radius and height.

$\displaystyle \Pi r^{2} + \Pi r\sqrt{r^{2} + h^{2}}= \Pi\ast 4^{2} + \Pi \ast 4\sqrt{4^{2} + 3^{2}} = 16\Pi + 4\Pi \sqrt{25} = 16\Pi + 20\Pi = 36\Pi$

Lateral Area = LA = π(r)(l) where r = radius of the base and l = slant height

LA = 2B

π(r)(l) = 2π(r²)

rl = 2r²

l = 2r

From the diagram, we can see that r² + h² = l².  Since h = 9 and l = 2r, some substitution yields

r² + 9² = (2r)² 

r² + 81 = 4r² 

81 = 3r² 

27 = r²

B = π(r²) = 27π

LA = 2B = 2(27π) = 54π

SA = B + LA = 81π

To find the surface area of a cone we must plug in the appropriate numbers into the equation

$\displaystyle Surface\ Area=\pi r^{2}+\pi rl$

where $\displaystyle r$ is the radius of the base, and $\displaystyle l$ is the lateral, or slant height of the cone.

First we must find the area of the circle.

To find the area of the circle we plug in our radius into the equation of a circle which is

This yields $\displaystyle A=25\pi$.

We then need to know the surface area of the cone shape.

To find this we must use our height and our radius to make a right triangle in order to find the lateral height using Pythagorean’s Theorem.

Pythagorean’s Theorem states

Take the radius and height and plug them into the equation as a and b to yield $\displaystyle 5^{2}+8^{2}=c^{2}$

First square the numbers $\displaystyle 25+64=c^{2}$

After squaring the numbers add them together $\displaystyle 89=c^{2}$

Once you have the sum, square root both sides $\displaystyle \sqrt{89}=\sqrt{c^{2}}$

After calculating we find our length is $\displaystyle c=l=9.43$

Then plug the length into the second portion of our surface area equation above to get $\displaystyle (4)(\pi)(9.43)=37.72\pi$

Then add the area of the circle with the conical area to find the surface area of the entire figure $\displaystyle 25\pi+37.72\pi=62.72\pi$

The answer is $\displaystyle 62.72\pi$.

Find the slant height of the cone using the Pythagorean theorem:  r² + h² = s² resulting in 6² + 8² = s² leading to s² = 100 or s = 10 in

SA = πrs + πr² = π(6)(10) + π(6)² = 60π + 36π = 96π in²

60π in² is the area of the cone without the base.

36π in² is the area of the base only.

The standard equation to find the surface area of a cone is 

$\displaystyle SA=\pi rs+\pi r^2$

where $\displaystyle s$ denotes the slant height of the cone, and $\displaystyle r$ denotes the radius.

Plug in the given values for $\displaystyle s$ and $\displaystyle r$ to find the answer:

$\displaystyle SA=\pi (12)(15)+\pi (12)^2=180\pi +144\pi =324\pi$

The formula for the surface area of a cone is:

$\displaystyle SA = A_{base} + A_{lateral}$

$\displaystyle SA = \pi r^2 + \pi r l$

where $\displaystyle r$ is the radius of the cone and $\displaystyle l$ is the slant height of the cone.

Plugging in our values, we get:

$\displaystyle SA = \pi (5m)^2 + \pi (5m) (13m)$

$\displaystyle SA = 25 \pi m^2 + 65 \pi m^2 = 90 \pi m^2$

The formula for the surface area of a cone is:

$\displaystyle SA = (base) + \frac{(circumference)(slant height)}{2}$

$\displaystyle SA = (\pi r^2) + \frac{(2 \pi r)(h_s)}{2}$

$\displaystyle SA = (\pi r^2) + ( \pi rh_s)$

Use the Pythagorean Theorem to find the length of the radius:

$\displaystyle A^2 + B^2 = C^2$

$\displaystyle A^2 + (8m)^2 = (10m)^2$

$\displaystyle A = 6m$

Plugging in our values, we get:

$\displaystyle SA = \pi (6m)^2 + \pi (6m)(10m)$

$\displaystyle SA = 96 \pi m^2$

The formula for the surface area of the half cone is:

$\displaystyle SA=(base)+\frac{1}{2}(circumference)(slant\ height)+(triangle)$

$\displaystyle SA=\left(\frac{\pi r^2}{2}\right)+\left(\frac{1}{2}\cdot \frac{2\pi r h_s}{2}\right)+rh$

Where $\displaystyle r$ is the radius, $\displaystyle h_s$ is the slant height, and $\displaystyle h$ is the height of the cone.

Use the Pythagorean Theorem to find the height of the cone:

$\displaystyle A^2+B^2 = C^2$

$\displaystyle A^2+(8m)^2=(17m)^2$

$\displaystyle A=15m$

Plugging in our values, we get:

$\displaystyle SA=\left(\frac{\pi (8m)^2}{2}\right)+\left( \frac{\pi (8m) (17m)}{2}\right)+(8m)(15m)$

$\displaystyle SA=32 \pi m^2 + 68 \pi m^2 + 120m^2$

$\displaystyle SA = 100 \pi m^2 + 120 m^2$