the base of a right pyramid is a regular hexagon with sides of length 8 m. the altitude is 4 m. find the total surface area of the pyramid. the area is (simplify your ar exact answer, using radicals as needed)

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Accepted Answer

# Explanation:
## Step1: Calculate the area of the base
The formula for the area of a regular hexagon is $A_{base}=\frac{3\sqrt{3}}{2}s^{2}$, where $s = 8$ m.
$$
\begin{align*}
A_{base}&=\frac{3\sqrt{3}}{2}\times8^{2}\
&=\frac{3\sqrt{3}}{2}\times64\
&=96\sqrt{3}\text{ m}^2
\end{align*}
$$
## Step2: Calculate the slant height ($l$)
We know that the altitude ($h = 4$ m) and half - side of the hexagon ($a=\frac{8}{2}=4$ m). Using the Pythagorean theorem $l=\sqrt{h^{2}+a^{2}}$
$$
\begin{align*}
l&=\sqrt{4^{2}+4^{2}}\
&=\sqrt{16 + 16}\
&=\sqrt{32}\
&=4\sqrt{2}\text{ m}
\end{align*}
$$
## Step3: Calculate the lateral surface area ($A_{lateral}$)
The lateral surface area of a regular pyramid is $A_{lateral}=\frac{1}{2}Pl$, where $P$ is the perimeter of the base. For a regular hexagon with side $s = 8$ m, $P=6s=6\times8 = 48$ m and $l = 4\sqrt{2}$ m
$$
\begin{align*}
A_{lateral}&=\frac{1}{2}\times48\times4\sqrt{2}\
&=24\times4\sqrt{2}\
&=96\sqrt{2}\text{ m}^2
\end{align*}
$$
## Step4: Calculate the total surface area ($A_{total}$)
The total surface area of the pyramid is $A_{total}=A_{base}+A_{lateral}$
$$
\begin{align*}
A_{total}&=96\sqrt{3}+96\sqrt{2}\
&=96(\sqrt{3}+\sqrt{2})\text{ m}^2
\end{align*}
$$

# Answer:
$96(\sqrt{3}+\sqrt{2})\text{ m}^2$

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QUESTION IMAGE

Question

Explanation:

Step1: Calculate the area of the base

Step2: Calculate the slant height (\(l\))

Step3: Calculate the lateral surface area (\(A_{lateral}\))

Step4: Calculate the total surface area (\(A_{total}\))

Answer: