Question
Find the volume of the square pyramid. Base is 5cm and slant height is 6cm
45.5
50
30
150
45.5
50
30
150
Answers
GPT-4o mini
To find the volume of a square pyramid, we can use the formula:
\[
V = \frac{1}{3} \times B \times h
\]
where \(B\) is the area of the base and \(h\) is the height of the pyramid.
The base of the pyramid is a square, so the area of the base \(B\) can be calculated as:
\[
B = \text{side}^2
\]
Given that the side length of the base is 5 cm:
\[
B = 5^2 = 25 \text{ cm}^2
\]
Next, we need to find the height \(h\) of the pyramid. We are given the slant height, which is 6 cm. To find the height, we can visualize the pyramid and create a right triangle where:
- One leg is the height \(h\),
- The other leg is half the base length (which is \( \frac{5}{2} = 2.5 \text{ cm} \)),
- The hypotenuse is the slant height (6 cm).
Using the Pythagorean theorem:
\[
h^2 + (2.5)^2 = 6^2
\]
Calculating \(6^2\):
\[
h^2 + 6.25 = 36
\]
\[
h^2 = 36 - 6.25
\]
\[
h^2 = 29.75
\]
\[
h = \sqrt{29.75} \approx 5.45 \text{ cm}
\]
Now we can substitute \(B\) and \(h\) back into the volume formula:
\[
V = \frac{1}{3} \times 25 \times 5.45
\]
Calculating:
\[
V = \frac{1}{3} \times 136.25 \approx 45.42 \text{ cm}^3
\]
Rounding this gives approximately \(45.5 \text{ cm}^3\).
Thus, the volume of the square pyramid is approximately \( \boxed{45.5} \text{ cm}^3\).
\[
V = \frac{1}{3} \times B \times h
\]
where \(B\) is the area of the base and \(h\) is the height of the pyramid.
The base of the pyramid is a square, so the area of the base \(B\) can be calculated as:
\[
B = \text{side}^2
\]
Given that the side length of the base is 5 cm:
\[
B = 5^2 = 25 \text{ cm}^2
\]
Next, we need to find the height \(h\) of the pyramid. We are given the slant height, which is 6 cm. To find the height, we can visualize the pyramid and create a right triangle where:
- One leg is the height \(h\),
- The other leg is half the base length (which is \( \frac{5}{2} = 2.5 \text{ cm} \)),
- The hypotenuse is the slant height (6 cm).
Using the Pythagorean theorem:
\[
h^2 + (2.5)^2 = 6^2
\]
Calculating \(6^2\):
\[
h^2 + 6.25 = 36
\]
\[
h^2 = 36 - 6.25
\]
\[
h^2 = 29.75
\]
\[
h = \sqrt{29.75} \approx 5.45 \text{ cm}
\]
Now we can substitute \(B\) and \(h\) back into the volume formula:
\[
V = \frac{1}{3} \times 25 \times 5.45
\]
Calculating:
\[
V = \frac{1}{3} \times 136.25 \approx 45.42 \text{ cm}^3
\]
Rounding this gives approximately \(45.5 \text{ cm}^3\).
Thus, the volume of the square pyramid is approximately \( \boxed{45.5} \text{ cm}^3\).