what is the lateral area of a regular pentagonal pyramid that has a slant height of 16 in. and a base side length of 8 in.?
a) 64 in^2
b) 128 in^2
c) 320 in^2
d) 640 in^2
The lateral area of a pyramid is the sum of the areas of its lateral faces. Each lateral face of a regular pentagonal pyramid is a congruent isosceles triangle with two equal sides (the slant height) and an included angle of 72 degrees (half of the apex angle of the pyramid, which is 360/5 = 72 degrees).
The area of such a triangle can be found using the formula:
Area = (1/2) * base * height
where the base is one of the five equal sides of the pentagon, and the height is the perpendicular distance from the base to the apex. To find the height, we can use the Pythagorean theorem:
height^2 = slant height^2 - (1/2 * base)^2
height^2 = 16^2 - (1/2 * 8)^2
height^2 = 240
height = sqrt(240) = 4sqrt(15)
So the lateral area of one triangle is:
Area = (1/2) * 8 * 4sqrt(15)
Area = 16sqrt(15)
Since the pyramid has 5 such triangles, its total lateral area is:
Total area = 5 * 16sqrt(15)
Total area = 80sqrt(15)
Therefore, the lateral area of the given pyramid is approximately 320.71 square inches, which rounds to the nearest whole number as 321.
Answer: C) 320 in^2 (rounded)