Question

The area of the base and total surface area of a square based solid pyramid are 144 sq. cm and 384 sq cm respectively. What will be the volume of the pyramid? Find it​

Answer 1

Answer:

  384 cubic centimeters

Step-by-step 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.