The lateral edges of a regular hexagonal pyramid are all 20 cm long, and the base edges are all 16 cm long. To the nearest cc, what is the volume of this pyramid? To the nearest square cm, what is the combined area of the base and six lateral faces?
The base area equals six times the area of an equilateral triangle with side length 20 cm (since six equilateral triangles make up a hexagon). Multiply that area by the base edge height for the volume.
Add twice the base area to the area of the six 20x16 cm faces to get the area.
hint: equilateral triangle area =
(1/2)*(side length)^2*[sqrt3)/2]
To find the volume of the pyramid, we can use the formula:
Volume = (1/3) * Base Area * Height.
Since the pyramid is regular, its base is a regular hexagon. To find the area of a regular hexagon, we can use the formula:
Area = (3√3 * side^2) / 2.
Given that the base edges are all 16 cm long, we can substitute it into the formula to find the base area:
Base Area = (3√3 * 16^2) / 2.
Next, we need to find the height of the pyramid. To do that, we can use the Pythagorean theorem on one of the triangular faces. Considering a lateral edge and half of the base edge as the two legs, the hypotenuse (which is also the height) can be calculated as follows:
Height = √(edge^2 - (base/2)^2).
Substituting the given values, we have:
Height = √(20^2 - (16/2)^2).
Once we have the base area and the height, we can calculate the volume:
Volume = (1/3) * Base Area * Height.
Compute the numerical value of the volume using the given numbers to get the answer.
To find the combined area of the base and six lateral faces, we need to calculate the total surface area. The total surface area consists of the area of the base and the area of the six triangular lateral faces.
The area of the base is already calculated. To find the area of one lateral face, we can use the formula for the area of an equilateral triangle:
Lateral Face Area = (√3 / 4) * side^2.
We can substitute the given lateral edge length (20 cm) into the formula to find the area of each lateral face, and then multiply it by 6 to account for all six faces.
Finally, add the base area and the combined area of the lateral faces to get the total surface area. Round the answer to the nearest square cm to get the final result.