Question

A box of coffee is in the shape of a hexagonal prism. Each of the 6 triangles has the same dimension (all triangles are the same).

Calculate the SURFACE AREA ( outside of this shape not including the inside) to determine how much cardboard is needed to construct this packaging.

Answer 1

429.9

Explanation:

Answer 2

The total surface area of the hexagonal prism is 429.9

To calculate the surface area of a hexagonal prism, we need to calculate the area of the two hexagonal bases and the area of the six rectangular sides.

1. Calculate the area of one hexagonal base: A regular hexagon can be divided into six equilateral triangles. The area of an equilateral triangle with side length s is given by the formula
`\( A = (√(3))/(4) s^2 \).`Since we have a hexagon made up of six equilateral triangles, we multiply the area of one triangle by six.

2. Calculate the area of the six rectangular sides: Each rectangular side has one dimension that is the side length of the hexagon and the other dimension is the height of the prism. The area of each rectangle is simply
`\( length * width \).`

Given the dimensions in the diagram:

Side length of hexagon s = 5 cm

Height of the prism h = 10 cm

Let's perform the calculations:

Hexagonal Base Area:

`\[ \text{Base area} = 6 * \left( (√(3))/(4) s^2 \right) \]`

Rectangular Side Area:

`\[ \text{Rectangular side area} = s * h \]`

`\[ \text{Total area of sides} = 6 * (\text{Rectangular side area}) \]`

Total Surface Area:

`\[ \text{Total surface area} = (\text{Base area}) * 2 + (\text{Total area of sides}) \]`

The total surface area = 429.9 square centimeters when rounded to the nearest hundredth.