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Sagot :
To determine the total area of a Pyramid you have to use the following formula:
[tex]TA=A_{\text{base}}+\frac{1}{2}(P_{\text{base}}\cdot s)[/tex]Where
A_base refers to the area of the base, in this case, it would be the area of the hexagon
P_base refers to the perimeter of the base, in this case, the perimeter of the hexagon
s indicates the slant height of the pyramid
Before you can determine the total area of the pyramid, you have to calculate the area and perimeter of the regular hexagon.
The perimeter of the hexagon
To determine the perimeter of any shape, you have to add the length of its sides, in this case, the hexagon is regular which means that all sides are equal, so the perimeter is equal to 6 times the side length (a):
For a=4
[tex]\begin{gathered} P_{\text{base}}=6a \\ P_{\text{base}}=6\cdot4 \\ P_{\text{base}}=24\text{units} \end{gathered}[/tex]The area of the hexagon
To determine this area you have to use the following formula:
[tex]A_{\text{base}}=\frac{3\sqrt[]{3}a^2}{2}[/tex]Where "a" represents the side length
For a=4, the area of the base is:
[tex]\begin{gathered} A_{\text{base}}=\frac{3\sqrt[]{3\cdot}4^2}{2} \\ A_{\text{base}}=\frac{3\sqrt[]{3}\cdot16}{2} \\ A_{\text{base}}=24\sqrt[]{3}\text{units}^2 \end{gathered}[/tex]Total area of the pyramid
Now that we determined the area and perimeter of the base, given that the slant height of the pyramid is s=6, we can calculate the total area as follows:
[tex]\begin{gathered} TA=A_{\text{base}}+\frac{1}{2}(P_{\text{base}}\cdot s) \\ TA=24\sqrt[]{3}+\frac{1}{2}(24\cdot6) \\ TA=24\sqrt[]{3}+\frac{1}{2}\cdot144 \\ TA=24\sqrt[]{3}+72 \end{gathered}[/tex]The total area of the hexagonal pyramid is 72+24√3 units²
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