4. Find the lateral area of a regular hexagonal pyramid if each base edge is 12 cm and each lateral edge is 10 cm. Round to the nearest tenth.
To find the lateral area of a regular hexagonal pyramid, we first need to find the apothem (height) of the lateral face.
In a regular hexagon, the apothem can be calculated using the formula:
apothem = (side length) / 2 * tan(30 degrees)
Since the base edge of the pyramid is 12 cm, the side length of the hexagon is also 12 cm.
apothem = (12/2) * tan(30)
apothem = 6 * tan(30)
apothem = 6 * 0.5774
apothem = 3.4644 cm
Next, we need to find the perimeter of the base of the pyramid, which is a regular hexagon.
Perimeter = 6 * base edge
Perimeter = 6 * 12
Perimeter = 72 cm
Now we can calculate the lateral area of the pyramid using the formula:
Lateral area = (perimeter * lateral edge) / 2
Lateral area = (72 * 10) / 2
Lateral area = 720 / 2
Lateral area = 360 cm^2
Therefore, the lateral area of the regular hexagonal pyramid is 360 cm^2.