To solve for the total surface area of a regular hexagonal pyramid with an edge length of 8 cm at the base and an altitude of 12 cm, follow these steps:
1. Calculate the Area of the Base:
The base of the pyramid is a regular hexagon. The formula to find the area of a regular hexagon with side length [tex]\( a \)[/tex] is:
[tex]\[
\text{Area of the base} = \frac{3 \sqrt{3}}{2} a^2
\][/tex]
Substituting [tex]\( a = 8 \)[/tex] cm:
[tex]\[
\text{Area of the base} = \frac{3 \sqrt{3}}{2} (8^2) \approx 166.2769 \text{ cm}^2
\][/tex]
2. Calculate the Slant Height of the Pyramid:
To find the slant height, we need to use the Pythagorean theorem in the right triangle formed by the altitude, the radius of the base, and the slant height. The radius of the base ([tex]\( r \)[/tex]) can be found using:
[tex]\[
r = a \frac{\sqrt{3}}{2}
\][/tex]
Substituting [tex]\( a = 8 \)[/tex] cm:
[tex]\[
r = 8 \times \frac{\sqrt{3}}{2} \approx 6.9282 \text{ cm}
\][/tex]
Now, applying the Pythagorean theorem to find the slant height ([tex]\( l \)[/tex]):
[tex]\[
l = \sqrt{r^2 + \text{altitude}^2} = \sqrt{6.9282^2 + 12^2} \approx 13.8564 \text{ cm}
\][/tex]
3. Calculate the Lateral Surface Area:
The lateral surface area consists of 6 isosceles triangles. Each of these triangles has a base of 8 cm (the edge of the hexagon) and a slant height [tex]\( l \)[/tex]. The area of one triangle is given by:
[tex]\[
\text{Area of one triangle} = \frac{1}{2} \times \text{base} \times l
\][/tex]
So the lateral surface area is:
[tex]\[
\text{Lateral surface area} = 6 \times \frac{1}{2} \times 8 \times 13.8564 \approx 332.5538 \text{ cm}^2
\][/tex]
4. Calculate the Total Surface Area:
The total surface area is the sum of the base area and the lateral surface area:
[tex]\[
\text{Total surface area} = \text{Area of the base} + \text{Lateral surface area} \approx 166.2769 + 332.5538 \approx 498.8306 \text{ cm}^2
\][/tex]
Thus, the total surface area of the regular hexagonal pyramid is approximately 498.831 cm².
