Answer:
384 cubic centimeters
Explanation:
We are given the base area and total area of a square pyramid and asked to find the volume. The typical volume formula for a pyramid requires that we know the height of it. We can use the formula for the surface area to find the slant height of a face, and we can use the slant height to find the height perpendicular to the base.
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face area
total surface area = base area + area of 4 faces
area of 4 faces = total surface area - base area
area of 4 faces = 384 cm² -144 cm² = 240 cm²
area of 1 face = 240 cm²/4 = 60 cm²
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slant height
The side length of the square base is ...
A = a²
a = √A = √(144 cm²) = 12 cm
and the slant height of one face is ...
A = 1/2as
s = (2A)/a = (2×60 cm²)/(12 cm) = 10 cm
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pyramid height
The distance from the center of the base to the edge of the base is (12 cm)/2 = 6 cm. That is one leg of the right triangle whose hypotenuse is the slant height of the face. The other leg of the triangle is the height of the pyramid:
H² = s² -(a/2)² . . . . Pythagorean theroem solved for one leg
H = √(10² -6²) = 8 . . . . cm . . . . height of the pyramid
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volume
Now we know the base area of the pyramid (B), and we know its height (H), so we can use the formula ...
V = 1/3BH
V = 1/3(144 cm²)(8 cm) = 384 cm³
The volume of the pyramid will be 384 cm³.