what volume of a square pyramid with base edges of 24 cm and a slant height of 37 cm?
a) 7104 cm^3
b) 20160 cm^3
c) 10080 cm^3
d) 6720 cm^3
The formula for the volume of a square pyramid is (1/3) x (base area) x height.
First, let's find the height of the pyramid using the Pythagorean theorem:
height^2 = slant height^2 - (1/2 x base edge)^2
height^2 = 37^2 - (24/2)^2
height^2 = 1369 - 288
height^2 = 1081
height = 32.91 cm
Now, let's find the area of the base:
base area = (side length)^2
base area = 24^2
base area = 576 cm^2
Finally, we can calculate the volume:
volume = (1/3) x (base area) x height
volume = (1/3) x 576 x 32.91
volume = 10080 cm^3
Therefore, the answer is c) 10080 cm^3.
To find the volume of a square pyramid, you can use the formula:
Volume = (1/3) * (base area) * (height)
The base area can be found by squaring one of the base edges:
Base area = (24 cm)^2 = 576 cm^2
The height can be found using the Pythagorean theorem. The height, slant height, and half of a base edge form a right triangle. The height is the leg of this triangle.
Using the Pythagorean theorem:
(height)^2 + (half of base edge)^2 = (slant height)^2
Let's solve for the height:
(height)^2 + (12 cm)^2 = (37 cm)^2
(height)^2 + 144 cm^2 = 1369 cm^2
(height)^2 = 1369 cm^2 - 144 cm^2
(height)^2 = 1225 cm^2
Taking the square root of both sides, we get:
height = √(1225 cm^2)
height = 35 cm
Now, let's calculate the volume:
Volume = (1/3) * (base area) * (height)
Volume = (1/3) * (576 cm^2) * (35 cm)
Volume = 6720 cm^3
So, the volume of the square pyramid is 6720 cm^3, which corresponds to option d).