Before the shingles of the steeple (see Exercise 29) are replaced, an exhaust fan is to be installed in the steeple. To determine what size exhaust fan should be installed, it is necessary to know the volume of air in the steeple. Find the volume of the regular square pyramid described in Exercise 29.
The volume of the pyramid.
A regular square pyramid as below,
and lateral edge measures and each base edge measures .
A pyramid is made by connecting a base to an apex. The base is flat with straight edges, no curves, hence, a polygon and all other faces are triangles.
A regular pyramid is a pyramid whose base is a regular polygon and whose lateral edges are all congruent.
The lateral area of a regular pyramid with slant height of length and perimeter of the base is given by
The total surface area of a pyramid with lateral area and base area is given by
According to the Pythagorean theorem, in a right-angled triangle
In a regular pyramid, the lengths of the apothem of the base, the altitude , and the slant height satisfy the Pythagorean Theorem; that is, .
The perimeter of the square is four times the side.
The volume of a pyramid having a base area and an altitude of length is given by .
From the given figure
the sides of the square base measure each and the apothem is half the base length. Hence, . The lateral edge measures or .
The slant height
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