The perimeter of a regular hexagonal pyramid is 24 cm. The angle between the pyramid's slant height and hexagon's apothem is 60 degrees. find the volume of the pyramid
To find the volume of the pyramid, we need to know the height of the pyramid first.
Let's start by finding the length of one side of the hexagon base.
The perimeter of a regular hexagon is given by the formula: Perimeter = 6 * Side length.
Therefore, Side length = Perimeter / 6 = 24 cm / 6 = 4 cm.
Now, let's find the apothem of the hexagon.
The apothem of a regular hexagon is given by the formula: Apothem = Side length / 2 = 4 cm / 2 = 2 cm.
Next, we need to find the slant height of the pyramid.
A slant height is the height of each triangular face of the pyramid. In the case of a regular hexagonal pyramid, the slant height connects the vertices of the base to the apex of the pyramid.
We know that the angle between the slant height and the hexagon's apothem is 60 degrees. This forms a right triangle with the slant height being the hypotenuse.
Using trigonometry, we can find the slant height.
sin(60 degrees) = Opposite / Hypotenuse.
sin(60 degrees) = 2 cm / Hypotenuse.
Hypotenuse = 2 cm / sin(60 degrees).
Hypotenuse ≈ 2.309 cm (rounded to 3 decimal places).
Now that we have the slant height and the apothem, we can find the height of the pyramid.
Using the Pythagorean theorem: Height² = Slant height² - Apothem².
Height² = 2.309 cm² - 2 cm².
Height² ≈ 0.754 cm².
Taking the square root of both sides, Height ≈ 0.868 cm (rounded to 3 decimal places).
Finally, we can calculate the volume of the pyramid.
The volume of a pyramid is given by the formula: Volume = (1/3) * Base Area * Height.
The base area of a regular hexagon is given by the formula: Base Area = (3 * (√3) * Side length²) / 2.
Substituting the values, we have:
Base Area = (3 * (√3) * (4 cm)²) / 2 = (3 * (√3) * 16 cm²) / 2 = 24√3 cm² (approx.).
Volume = (1/3) * 24√3 cm² * 0.868 cm = 6.9√3 cm³ (approx.).
Therefore, the volume of the pyramid is approximately 6.9√3 cm³.
To find the volume of a pyramid, we need the base area and the height. In this case, since we have a regular hexagonal pyramid, we can calculate the base area using the given perimeter.
To find the base area of the hexagon, we first need to find the length of one side. Since a regular hexagon has all equal sides, we can divide the perimeter by the number of sides, which gives us:
Side length = Perimeter / Number of sides
Side length = 24 cm / 6
Side length = 4 cm
Now that we know the side length of the hexagon, we can find the apothem, which is the distance from the center of the hexagon to any of its sides. The apothem can be calculated using the formula:
Apothem = Side length / (2 * tan(π / Number of sides))
Apothem = 4 cm / (2 * tan(π / 6))
Apothem = 4 cm / (2 * tan(π / 6))
Apothem = 4 cm / (2 * tan(π / 6))
Apothem = 4 cm / (2 * tan(π / 6))
Apothem = 4 cm / (2 * 1.732)
Apothem = 4 cm / 3.464
Apothem ≈ 1.155 cm
Next, we need to find the slant height of the pyramid. The slant height is the distance from the apex (top) of the pyramid to any of the vertices on the base hexagon. Given that the angle between the slant height and the apothem is 60 degrees, we can use trigonometry to find the slant height.
Using the formula:
Slant height = Apothem / cos(Angle between slant height and apothem)
Slant height = 1.155 cm / cos(60 degrees)
Slant height = 1.155 cm / 0.5
Slant height = 2.31 cm
Now that we have the base area (hexagon) and the slant height, we can find the height of the pyramid. Using the formula:
Height = sqrt(Slant height^2 - Apothem^2)
Height = sqrt(2.31 cm^2 - 1.155 cm^2)
Height = sqrt(5.3361 cm^2 - 1.3339 cm^2)
Height = sqrt(4.0022 cm^2)
Height ≈ 2 cm
Finally, we can calculate the volume of the pyramid using the formula:
Volume = (1/3) * Base area * Height
Volume = (1/3) * (Area of hexagon) * Height
Volume = (1/3) * (Base area) * Height
Volume = (1/3) * ((3 * sqrt(3)/2) * Side length^2) * Height
Volume = (1/3) * ((3 * sqrt(3)/2) * 4 cm^2) * 2 cm
Volume = (1/3) * (6 * sqrt(3) * 4 cm^2) * 2 cm
Volume = (1/3) * (48 * sqrt(3) cm^2) * 2 cm
Volume = 32 * sqrt(3) cm^3
Therefore, the volume of the pyramid is approximately 55.42 cm^3.
Each side of the base is 6 cm ------- (24/6)
make a sketch of the hexagon base to find its apothem, the distance
from the centre to the midpoint of a side.
Consider one of the equilateral triangles of side 6
You should have no difficulty to find the apothem to be 3√3
you said the angle between slant and apothem is 60°
let the height of the pyramid be h cm
then tan60° = h/3√3
h = 3√3tan60 = 9
volume = (1/3)(base)(height) = (1/3)(base)(9) = 3base
I will leave it up to you to find the area of the hexagon