Two identical regular right pyramids, each with a square base of side 6 cm and a perpendicular height of 4 cm, are stacked one on top of the other such that their bases perfectly align. The top pyramid is inverted. Calculate the slant height of a single pyramid.
5 cm
Only one pyramid's slant height is needed; the stacking is background detail and does not change a single pyramid's dimensions.
In a square-based right pyramid the apex sits directly above the centre of the square, so the perpendicular height, the slant height and half of a base side form a right triangle.
Half the base side is \(\dfrac{a}{2} = \dfrac{6}{2} = 3\) cm, which is the horizontal leg (the apothem of the square base).
The perpendicular height is the vertical leg, \(h = 4\) cm.
The slant height \(l\) is the hypotenuse: \(l = \sqrt{h^2 + \left(\dfrac{a}{2}\right)^2} = \sqrt{4^2 + 3^2}\).
Evaluating, \(l = \sqrt{16 + 9} = \sqrt{25} = 5\) cm (the familiar 3-4-5 right triangle).
The slant height of a single pyramid is 5 cm.
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