Which expression represents the volume of the pyramid?
A solid right pyramid has a square base with an edge length of s units and a height of h units.
A. 1 s?h units
B. sh units
C. sh units
D. 3sh units
The Correct Answer and Explanation is:
The correct answer is C. (\frac{1}{3}sh) units.
Explanation:
A pyramid is a three-dimensional shape with a polygonal base and triangular faces that converge at a single point called the apex. In the case of a right pyramid with a square base, the following parameters are provided:
- Edge length of the base: (s) (units).
- Height: (h) (units), which is the perpendicular distance from the base to the apex.
The volume (V) of any pyramid can be calculated using the general formula:
[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
]
For a square base, the area of the base (A) is simply the square of the edge length, so:
[
A = s^2
]
Now, substitute this into the volume formula:
[
V = \frac{1}{3} \times s^2 \times h
]
This gives:
[
V = \frac{1}{3} s^2 h
]
Thus, the expression that represents the volume of the pyramid with a square base and a height of (h) is:
[
V = \frac{1}{3} s^2 h
]
This formula expresses the relationship between the base’s edge length ((s)), the height ((h)), and the volume of the pyramid.
Let’s break it down further:
- Base Area: The area of the square base is (s^2) because the area of a square is the square of its side length.
- Height: The height ((h)) is the perpendicular distance from the center of the square base to the apex.
- Scaling Factor: The factor (\frac{1}{3}) in the formula accounts for the fact that the volume of a pyramid is one-third of the volume of a prism with the same base area and height.
Thus, the correct formula for the volume of this pyramid is (\frac{1}{3} s^2 h), which matches option C.