Find the lateral surface area for the regular pentagon prism represented by the given net
5 ft 4 ft 6 ft
To find the lateral surface area of the regular pentagonal prism, we first need to find the area of each side of the prism.
The pentagon has 5 sides, each with a base of 4 ft and a height of 6 ft.
The formula for finding the area of a pentagon is:
Area = 1/2 x Perimeter x Apothem
First, we need to find the perimeter of the pentagon:
Perimeter = 5 x 4 = 20 ft
Next, we need to find the apothem of the pentagon.
To do this, we can divide the pentagon into 5 congruent triangles, with the apothem as the height of the triangle.
Using the Pythagorean theorem:
a^2 + (4/2)^2 = 6^2
a^2 + 2^2 = 6^2
a^2 + 4 = 36
a^2 = 32
a = √32 = 4√2 ft
Now we can find the area of one side of the prism:
Area = 1/2 x 20 ft x 4√2 ft
Area = 40√2 ft^2
Since there are 5 sides to the pentagon prism, the total lateral surface area of the prism is:
Total Area = 5 x 40√2 ft^2
Total Area ≈ 282.84 ft^2
Therefore, the lateral surface area of the regular pentagon prism is approximately 282.84 square feet.