THE BASE OF A REGULAR PYRAMID IS A REGULAR HEXAGON, EACH OF WHOSE SIDE IS 6 cm. IF THE SLANT HEIGHT OF THE PYRAMID IS 12 cm, CALCULATE THE LATERAL AREA, THE TOTAL SURFACE AREA, AND THE VOLUME OF THE PYRAMID  THE BASE OF A REGULAR PYRAMID IS A REGULAR HEXAGON, EACH OF WHOSE SIDE IS 6 cm. IF THE SLANT HEIGHT OF THE PYRAMID IS 12 cm, CALCULATE THE LATERAL AREA, THE TOTAL SURFACE AREA, AND THE VOLUME OF THE PYRAMID ÂÂ
The correct answer and explanation is :
To solve this problem, we need to calculate the lateral area, total surface area, and volume of a regular pyramid with a hexagonal base.
Given:
- Side length of the regular hexagon, ( s = 6 \, \text{cm} )
- Slant height of the pyramid, ( l = 12 \, \text{cm} )
Step 1: Lateral Area (LA)
The lateral area of a pyramid is given by:
[
\text{Lateral Area} = \frac{1}{2} \times \text{Perimeter of Base} \times \text{Slant Height}
]
The perimeter of a regular hexagon is the sum of the lengths of all its sides. Since the hexagon has six sides, the perimeter ( P ) is:
[
P = 6 \times s = 6 \times 6 = 36 \, \text{cm}
]
Now, we can calculate the lateral area:
[
\text{Lateral Area} = \frac{1}{2} \times 36 \times 12 = 216 \, \text{cm}^2
]
Step 2: Total Surface Area (TSA)
The total surface area of a pyramid is the sum of the lateral area and the area of the base. The base is a regular hexagon, and its area ( A_{\text{base}} ) is given by the formula:
[
A_{\text{base}} = \frac{3\sqrt{3}}{2} \times s^2
]
Substituting the value of ( s = 6 ):
[
A_{\text{base}} = \frac{3\sqrt{3}}{2} \times 6^2 = \frac{3\sqrt{3}}{2} \times 36 = 54\sqrt{3} \, \text{cm}^2
]
Approximating ( \sqrt{3} \approx 1.732 ):
[
A_{\text{base}} \approx 54 \times 1.732 = 93.648 \, \text{cm}^2
]
Now, we can calculate the total surface area:
[
\text{Total Surface Area} = \text{Lateral Area} + A_{\text{base}} = 216 + 93.648 \approx 309.648 \, \text{cm}^2
]
Step 3: Volume (V)
The volume of the pyramid is given by:
[
V = \frac{1}{3} \times A_{\text{base}} \times h
]
where ( h ) is the height of the pyramid. To find ( h ), we use the Pythagorean theorem. The height ( h ), slant height ( l ), and the radius ( r ) of the hexagonal base form a right triangle. The radius of a regular hexagon is equal to the side length ( r = s = 6 \, \text{cm} ).
Using the Pythagorean theorem:
[
l^2 = r^2 + h^2
]
Substitute the values of ( l ) and ( r ):
[
12^2 = 6^2 + h^2
]
[
144 = 36 + h^2
]
[
h^2 = 108
]
[
h = \sqrt{108} \approx 10.39 \, \text{cm}
]
Now, we can calculate the volume:
[
V = \frac{1}{3} \times 93.648 \times 10.39 \approx 324.6 \, \text{cm}^3
]
Final Answers:
- Lateral Area: 216 cm²
- Total Surface Area: 309.65 cm²
- Volume: 324.6 cm³
Explanation:
We first calculated the lateral area by using the formula for the lateral area of a pyramid, which depends on the perimeter of the base and the slant height. The total surface area was found by adding the lateral area and the area of the hexagonal base. Finally, the volume was calculated by determining the height of the pyramid using the Pythagorean theorem and applying the formula for the volume of a pyramid. Each calculation relies on geometric formulas that describe the relationships between the sides, slant height, and height of the pyramid.