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All the edges of a regular square pyramid have length 8.
A) What is the volume of the pyramid?
B) What is the lateral area of the pyramid?
C) What is the total surface area of the pyramid?

Sagot :

sqdancefan

Answer:

  A) (256/3)√2 ≈ 120.68 units³

  B) 64√3 ≈ 110.85 units²

  C 64(1+√3) ≈ 174.85 units²

Step-by-step explanation:

The dimensions used in the usual formulas for area and volume are not given, so it can work reasonably well to start by finding them.

Each face is an equilateral triangle with side length 8, so the slant height is the altitude of that triangle: 8(√3/2) = 4√3.

The height of the pyramid is the vertical leg of a right triangle with hypotenuse equal to the slant height (4√3) and horizontal leg equal to half the base side length. The height is found from the Pythagorean theorem:

  a² +b² = c²

  b = √(c² -a²) = √((4√3)² -4²) = 4√2

The pyramid with all sides of length 8 has height 4√2 and slant height 4√3.

__

A)

The volume is ...

  V = 1/3s²h . . . . . . where s is the side length, and h is the height

  V = 1/3(8²)(4√2) = (256/3)√2 ≈ 120.68 . . . . cubic units

__

B)

The lateral area is the total area of the 4 triangular faces, so is ...

  LA = 4(1/2)(bh) = 2(8)(4√3) = 64√3 ≈ 110.85 . . . . square units

h is the slant height in this formula

__

C)

The total surface area is the sum of the base area and the lateral area. The base area is the square of the side length.

  SA = 8² +64√3 = 64(1+√3) ≈ 174.85 . . . . square units

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