To find the surface area and volume of a pentagonal pyramid with an apothem of 3√2 and a height of 3, we need to use several formulas.
The apothem of a regular pentagon is the distance from the center of the pentagon to the midpoint of one of its sides. The area of a regular pentagon with side length s and apothem a is given by the formula:
A = (5/2) × s × a
The volume of a pyramid is given by the formula:
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 surface area of the pentagonal pyramid, we need to find the area of each face and add them together. Each face of the pentagonal pyramid is a triangle with one side as the base of the pyramid and the other two sides as the slant height of the pyramid. The slant height 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
The base of the pyramid is a regular pentagon, so we can find its area using the formula:
A = (5/2) × s × a
where s is 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 apothem and 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)
Therefore, the base area of the pyramid is:
A = (5/2) × s × a
A = (5/2) × 6√(5-2√5) × 3√2
A ≈ 31.18
To find the surface area of the 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).
Substituting the given values, 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
To find the volume of the pyramid, we can use the formula:
V = (1/3) × A × h
Substituting the given values, we get:
V = (1/3) × 31.18 × 3
V ≈ 31.18
Therefore, the surface area of the pentagonal pyramid is approximately 134.13 square units, and the volume of the pyramid is approximately 31.18 cubic units.