The volume and surface area of a pentagonal pyramid with an apothem of 3√2 and a height of 3 are:
Volume:
The formula for the volume of a pyramid is:
V = (1/3) × A × h
where A is the base area of the pyramid and h is the height of the pyramid.
To find the base area of the pentagonal pyramid, we need to find the length of one side of the pentagon. The apothem of the pentagon is given as 3√2, so we can find the length of one side using the formula for the apothem of a regular pentagon:
a = s / (2√(5-2√5))
3√2 = s / (2√(5-2√5))
s = 6√(5-2√5)
The base area of the pyramid is the area of a regular pentagon with side length s and apothem 3√2, which is given by the formula:
A = (5/2) × s × a
Substituting the given values, we get:
A = (5/2) × 6√(5-2√5) × 3√2
A ≈ 31.18
Therefore, the volume of the pentagonal pyramid is:
V = (1/3) × A × h
V = (1/3) × 31.18 × 3
V ≈ 31.18 cubic units
Surface Area:
To find the surface area of the pentagonal pyramid, we need to find the area of each of the five triangular faces. Each face has the same base area A and the same slant height l, so we can use the formula for the area of a triangle:
A = (1/2) × b × h
where b is the length of the base and h is the height of the triangle (which is the slant height of the pyramid).
The slant height of the pentagonal pyramid can be found using the Pythagorean theorem:
l = √(h^2 + a^2)
where h is the height of the pyramid and a is the apothem of the base.
Substituting the given values, we get:
l = √(3^2 + (3√2)^2)
l = √(9 + 18)
l = √27
l = 3√3
Substituting the values of A and l into the formula for the area of a triangle, we get:
A = (1/2) × A × l
A = (1/2) × 31.18 × 3√3
A ≈ 26.83
Since there are five triangular faces, the total surface area of the pentagonal pyramid is:
S = 5 × A
S ≈ 134.13 square units
Therefore, the volume of the pentagonal pyramid is approximately 31.18 cubic units, and the surface area is approximately 134.13 square units.