Find the total surface area of the following pyramids: A regular triangular pyramid has base edges of length 30 and height 10.

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  985 square units

Step-by-step explanation:

In order to determine the area of the triangular faces of the pyramid, we need to know the slant height of each face. That can be found using the Pythagorean theorem, if we know the distance from the pyramid altitude line to the middle of one side of the base.

That distance will be 1/3 of the median length of the base equilateral triangle. The median of an equilateral triangle with edge length 'a' is (√3/2)a, so 1/3 that is a/(2√3). For a=30, this is 30/(2√3) = 5√3.

The height (h) is given as 10, so the slant height s is ...

  s² = (5√3)² + 10² = 175

  s = √175 = 5√7 . . . . . . pyramid slant height

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The are three triangular faces, so their total area is ...

  3A = 3(1/2)(bh) = (3/2)(30)(5√7) = 225√7 . . . . . square units

The area of the triangular base is ...

  A = 1/2bh = 1/2(30)(15√3) = 225√3 . . . . . square units

Then the total surface area of the triangular pyramid is ...

  SA = base area + lateral area

  SA = 225√3 +225√7 ≈ 225(√3 +√7) ≈ 985.0 . . . square units

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