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Sagot :
To solve for the volume of the oblique pyramid, we need to follow these steps:
1. Calculate the Perimeter of the Hexagonal Base:
- The base is a regular hexagon, and each side (or edge) of the hexagon has a length of 8 cm.
- A regular hexagon has 6 sides.
- Therefore, the perimeter of the base is given by:
[tex]\[ \text{Perimeter} = 6 \times \text{side length} = 6 \times 8 = 48 \text{ cm} \][/tex]
2. Calculate the Area of the Hexagonal Base:
- It is given that the area of the hexagonal base is [tex]\(54 \sqrt{3} \text{ cm}^2\)[/tex].
- No additional calculation is needed here since the area is already provided:
3. Calculate the Height of the Pyramid:
- We use the information given about the angle BAC, which is [tex]\(60^\circ\)[/tex].
- To find the height [tex]\( h \)[/tex] of the pyramid, we need to use trigonometry.
- We know that in a regular hexagon, the apothem (the perpendicular distance from the center to a side) can be calculated using the formula for the apothem of a hexagon:
[tex]\[ \text{Apothem} = \frac{\text{side length}}{2} \times \sqrt{3} = \frac{8}{2} \times \sqrt{3} = 4\sqrt{3} \text{ cm} \][/tex]
- Since the angle [tex]\( \angle BAC = 60^\circ \)[/tex], the vertical height [tex]\( h \)[/tex] of the pyramid can be calculated using basic trigonometry:
[tex]\[ h = \text{Apothem} \times \sin(60^\circ) = 4\sqrt{3} \times \frac{\sqrt{3}}{2} = 4 \times \frac{3}{2} = 6 \text{ cm} \][/tex]
4. Calculate the Volume of the Pyramid:
- The volume [tex]\( V \)[/tex] of a pyramid is given by the formula:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
- Substituting the base area [tex]\(54 \sqrt{3} \text{ cm}^2\)[/tex] and the height [tex]\(6 \text{ cm}\)[/tex] into the formula, we get:
[tex]\[ V = \frac{1}{3} \times 54 \sqrt{3} \times 6 = \frac{1}{3} \times 324 \sqrt{3} = 108 \sqrt{3} \text{ cm}^3 \][/tex]
Hence, the volume of the pyramid is [tex]\(108 \sqrt{3} \text{ cm}^3\)[/tex]. Therefore, the correct answer is:
[tex]\[ 108 \sqrt{3} \text{ cm}^3 \][/tex]
1. Calculate the Perimeter of the Hexagonal Base:
- The base is a regular hexagon, and each side (or edge) of the hexagon has a length of 8 cm.
- A regular hexagon has 6 sides.
- Therefore, the perimeter of the base is given by:
[tex]\[ \text{Perimeter} = 6 \times \text{side length} = 6 \times 8 = 48 \text{ cm} \][/tex]
2. Calculate the Area of the Hexagonal Base:
- It is given that the area of the hexagonal base is [tex]\(54 \sqrt{3} \text{ cm}^2\)[/tex].
- No additional calculation is needed here since the area is already provided:
3. Calculate the Height of the Pyramid:
- We use the information given about the angle BAC, which is [tex]\(60^\circ\)[/tex].
- To find the height [tex]\( h \)[/tex] of the pyramid, we need to use trigonometry.
- We know that in a regular hexagon, the apothem (the perpendicular distance from the center to a side) can be calculated using the formula for the apothem of a hexagon:
[tex]\[ \text{Apothem} = \frac{\text{side length}}{2} \times \sqrt{3} = \frac{8}{2} \times \sqrt{3} = 4\sqrt{3} \text{ cm} \][/tex]
- Since the angle [tex]\( \angle BAC = 60^\circ \)[/tex], the vertical height [tex]\( h \)[/tex] of the pyramid can be calculated using basic trigonometry:
[tex]\[ h = \text{Apothem} \times \sin(60^\circ) = 4\sqrt{3} \times \frac{\sqrt{3}}{2} = 4 \times \frac{3}{2} = 6 \text{ cm} \][/tex]
4. Calculate the Volume of the Pyramid:
- The volume [tex]\( V \)[/tex] of a pyramid is given by the formula:
[tex]\[ V = \frac{1}{3} \times \text{Base Area} \times \text{Height} \][/tex]
- Substituting the base area [tex]\(54 \sqrt{3} \text{ cm}^2\)[/tex] and the height [tex]\(6 \text{ cm}\)[/tex] into the formula, we get:
[tex]\[ V = \frac{1}{3} \times 54 \sqrt{3} \times 6 = \frac{1}{3} \times 324 \sqrt{3} = 108 \sqrt{3} \text{ cm}^3 \][/tex]
Hence, the volume of the pyramid is [tex]\(108 \sqrt{3} \text{ cm}^3\)[/tex]. Therefore, the correct answer is:
[tex]\[ 108 \sqrt{3} \text{ cm}^3 \][/tex]
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