Question
Find the volume of the square pyramid (units are in cm)
Slanted height: 13, 10 underneath
Slanted height: 13, 10 underneath
Answers
GPT-4o mini
To find the volume of a square pyramid, we use the formula:
\[
V = \frac{1}{3} \times B \times h
\]
where \( V \) is the volume, \( B \) is the area of the base, and \( h \) is the height of the pyramid.
In this case, the base is a square with a side length of 10 cm. The area of the base \( B \) can be calculated as follows:
\[
B = \text{side}^2 = 10^2 = 100 \, \text{cm}^2
\]
Next, we need to find the vertical height \( h \) of the pyramid. We are given the slant height \( l = 13 \, \text{cm} \) and the side length of the base \( s = 10 \, \text{cm} \).
To find the vertical height \( h \), we can use the right triangle formed by the vertical height, half the base length, and the slant height:
- The half base length is \( \frac{s}{2} = \frac{10}{2} = 5 \, \text{cm} \).
Using the Pythagorean theorem in this triangle, we have:
\[
l^2 = h^2 + \left(\frac{s}{2}\right)^2
\]
Substituting the known values:
\[
13^2 = h^2 + 5^2
\]
Calculating the squares:
\[
169 = h^2 + 25
\]
Now, solving for \( h^2 \):
\[
h^2 = 169 - 25 = 144
\]
\[
h = \sqrt{144} = 12 \, \text{cm}
\]
Now that we have both the area of the base and the height, we can find the volume:
\[
V = \frac{1}{3} \times 100 \, \text{cm}^2 \times 12 \, \text{cm}
\]
Calculating the volume:
\[
V = \frac{1}{3} \times 1200 = 400 \, \text{cm}^3
\]
Thus, the volume of the square pyramid is:
\[
\boxed{400} \, \text{cm}^3
\]
\[
V = \frac{1}{3} \times B \times h
\]
where \( V \) is the volume, \( B \) is the area of the base, and \( h \) is the height of the pyramid.
In this case, the base is a square with a side length of 10 cm. The area of the base \( B \) can be calculated as follows:
\[
B = \text{side}^2 = 10^2 = 100 \, \text{cm}^2
\]
Next, we need to find the vertical height \( h \) of the pyramid. We are given the slant height \( l = 13 \, \text{cm} \) and the side length of the base \( s = 10 \, \text{cm} \).
To find the vertical height \( h \), we can use the right triangle formed by the vertical height, half the base length, and the slant height:
- The half base length is \( \frac{s}{2} = \frac{10}{2} = 5 \, \text{cm} \).
Using the Pythagorean theorem in this triangle, we have:
\[
l^2 = h^2 + \left(\frac{s}{2}\right)^2
\]
Substituting the known values:
\[
13^2 = h^2 + 5^2
\]
Calculating the squares:
\[
169 = h^2 + 25
\]
Now, solving for \( h^2 \):
\[
h^2 = 169 - 25 = 144
\]
\[
h = \sqrt{144} = 12 \, \text{cm}
\]
Now that we have both the area of the base and the height, we can find the volume:
\[
V = \frac{1}{3} \times 100 \, \text{cm}^2 \times 12 \, \text{cm}
\]
Calculating the volume:
\[
V = \frac{1}{3} \times 1200 = 400 \, \text{cm}^3
\]
Thus, the volume of the square pyramid is:
\[
\boxed{400} \, \text{cm}^3
\]