An unsharpened pencil is in the shape of a hexagonal prism, in which the face edge is 4 millimeters and the length is 170 millimeters. What is the lateral area of the pencil?
So you have 6 identical long skinny rectangles
each one is 4 by 170 mm^2 = 680 mm^2
multiply by 6
To find the lateral area of the pencil, we need to calculate the combined area of all the side faces.
First, let's calculate the perimeter of the hexagon. Since the face edge is 4 millimeters, the perimeter is 6 times that.
Perimeter of hexagon = 6 * 4 = 24 millimeters
Now, let's calculate the lateral area of each face. Since the shape is a hexagon, all the faces are identical. The area of each face is given by:
Area of hexagon face = apothem * perimeter / 2
To calculate the apothem, we can use the formula:
Apothem = (face edge) / (2 * tan(π/6))
Substituting the values:
Apothem = 4 / (2 * tan(π/6)) = 4 / (2 * tan(30°))
Using the approximate value for the tangent of 30 degrees (1/√3), we get:
Apothem = 4 / (2 * 1/√3) = 4 * √3 / 2 ≈ 3.46 millimeters
Now, we can calculate the area of each face:
Area of hexagon face = (3.46) * (24) / 2 ≈ 41.52 square millimeters
Since there are 2 identical side faces, the total lateral area of the pencil is:
Lateral area = 2 * (41.52) = 83.04 square millimeters
Therefore, the lateral area of the pencil is approximately 83.04 square millimeters.
To find the lateral area of the pencil, we need to calculate the surface area of the faces that form the sides of the hexagonal prism.
A hexagonal prism has two identical hexagonal bases and six rectangular faces. The lateral area is the sum of the areas of all six rectangular faces.
First, let's calculate the area of one rectangular face.
The length of the rectangular face is the same as the length of the pencil, which is 170 millimeters.
The width of the rectangular face is the same as the length of each edge of the hexagon, which is 4 millimeters.
So, the area of one rectangular face is calculated by multiplying the length and width.
Area of one rectangular face = Length × Width = 170 mm × 4 mm = 680 mm².
Since there are a total of six rectangular faces on the hexagonal prism, we need to multiply the area of one rectangular face by six to find the total lateral area of the pencil.
Total lateral area = Area of one rectangular face × 6 = 680 mm² × 6 = 4,080 mm².
Therefore, the lateral area of the pencil is 4,080 square millimeters.