Question
a rectangular pyramid has a square base with an area of 441 in.² and a height that is twice the length of a side of the base what is the volume of the pyramid?
Answers
GPT-4o mini
To find the volume of a rectangular pyramid (which, in this case, is a square pyramid since the base is a square), we will start by using the formula for the volume of a pyramid:
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Given that the area of the square base is 441 in², we can denote the side length of the base as \( s \). Since the area of a square is given by:
\[
s^2 = 441
\]
we can solve for \( s \):
\[
s = \sqrt{441} = 21 \text{ in}
\]
Next, we are told that the height \( h \) of the pyramid is twice the length of a side of the base:
\[
h = 2s = 2 \times 21 = 42 \text{ in}
\]
Now we can calculate the volume \( V \):
\[
V = \frac{1}{3} \times 441 \times 42
\]
Calculating the volume step by step:
1. Calculate \( 441 \times 42 \):
\[
441 \times 42 = 18522
\]
2. Now, divide by 3:
\[
V = \frac{18522}{3} = 6174 \text{ in}^3
\]
Therefore, the volume of the pyramid is
\[
\boxed{6174} \text{ in}^3.
\]
\[
V = \frac{1}{3} \times \text{Base Area} \times \text{Height}
\]
Given that the area of the square base is 441 in², we can denote the side length of the base as \( s \). Since the area of a square is given by:
\[
s^2 = 441
\]
we can solve for \( s \):
\[
s = \sqrt{441} = 21 \text{ in}
\]
Next, we are told that the height \( h \) of the pyramid is twice the length of a side of the base:
\[
h = 2s = 2 \times 21 = 42 \text{ in}
\]
Now we can calculate the volume \( V \):
\[
V = \frac{1}{3} \times 441 \times 42
\]
Calculating the volume step by step:
1. Calculate \( 441 \times 42 \):
\[
441 \times 42 = 18522
\]
2. Now, divide by 3:
\[
V = \frac{18522}{3} = 6174 \text{ in}^3
\]
Therefore, the volume of the pyramid is
\[
\boxed{6174} \text{ in}^3.
\]