A regular hexagonal pyramid has a perpendicular height of 42.0mm.If the distance across the flats of the hexagonal base is 18m find the volume and the lateral surface area of the pyramid.

the hexagon has an apothem a=9.

so, its sides are s=6√3

Now, you know how to find the area of a hexagon of side s.

The area A of the pyramid is just A=6(sa/2)

The volume is 1/3 (area of hexagon)*42

To find the volume and lateral surface area of the regular hexagonal pyramid, we can use the formulas:

- Volume = (1/3) * Base Area * Height
- Lateral Surface Area = Base Perimeter * Slant Height

Step 1: Find the Base Area
The base of a regular hexagon is made up of six equilateral triangles. To find the area of one of these triangles, we can divide it into two right-angled triangles.

The height of each right-angled triangle is half the distance across the flats (18m/2 = 9m), and the base is the length of one side of the hexagon (18m).

Using the formula for the area of a triangle, we can find the area of one of these triangles, then multiply it by 6 to find the base area of the hexagon.

Area of one triangle = (1/2) * base * height = (1/2) * 18m * 9m = 81m²
Base area of hexagon = 6 * Area of one triangle = 6 * 81m² = 486m²

Step 2: Find the Volume
We can plug the values of the base area (486m²) and the height (42.0mm) into the formula for the volume of a pyramid:

Volume = (1/3) * Base Area * Height
Volume = (1/3) * 486m² * 42.0mm

Since the units are different (m² and mm), we need to convert the height to meters.

1mm = 0.001m, so 42.0mm = 42.0 * 0.001m = 0.042m

Plugging in the values, we get:
Volume = (1/3) * 486m² * 0.042m = 27.216m³

Step 3: Find the Lateral Surface Area
To calculate the lateral surface area, we need to find the slant height. The slant height is the height of one of the triangular faces of the pyramid.

For an equilateral triangle with side length s, the height can be found using the Pythagorean theorem:
height = √(s² - (s/2)²) = √(s² - s²/4) = √(3s²/4)

In this case, s = 18m, so the slant height is:
slant height = √(3 * (18m)²/4) = √(3 * 324m²/4) = 18√3m

Now we can use the formula for the lateral surface area:
Lateral Surface Area = Base Perimeter * Slant Height

The perimeter of the hexagon is simply 6 times the length of one side, so the base perimeter is 6 * 18m = 108m.

Plugging in the values, we get:
Lateral Surface Area = 108m * 18√3m = 1944√3m²

Therefore, the volume of the regular hexagonal pyramid is 27.216m³ and the lateral surface area is 1944√3m².