Find the lateral surface area for the regular pentagonal prism represented by the given net
6 ft
4 ft
5 ft
The lateral surface area of a pentagonal prism is calculated by adding up the areas of all five faces.
Since the prism is regular, each face is a regular pentagon. The formula for the area of a regular pentagon is:
Area = (5/2) * a * s
where a is the apothem (distance from the center to the midpoint of a side) and s is the side length of the pentagon.
From the given dimensions, we can determine that the side length (s) is 4 ft and the apothem can be calculated using the Pythagorean theorem.
The apothem can be found by forming a right triangle with one leg being half the side length and the hypotenuse being the apothem.
a^2 + (s/2)^2 = 6^2
a^2 + 2^2 = 36
a^2 = 32
a = √32
a ≈ 5.66 ft
Now we can plug in the values into the formula for the area of a regular pentagon:
Area = (5/2) * 5.66 * 4
Area = (5/2) * 22.64
Area = 56.6 ft^2
Since there are five faces on the pentagonal prism, the total lateral surface area is 5 times the area of one face:
Total lateral surface area = 5 * 56.6
Total lateral surface area = 283 ft^2
Therefore, the lateral surface area of the regular pentagonal prism is 283 square feet.