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In geometry, the elongated square pyramid is a convex polyhedron constructed from a cube by attaching an equilateral square pyramid onto one of its faces. It is an example of Johnson solid.
ConstructionEdit
The elongated square pyramid is a composite, since it can constructed by attaching one equilateral square pyramid onto one of the faces of a cube, a process known as elongation of the pyramid.Template:R One square face of each parent body is thus hidden, leaving five squares and four equilateral triangles as faces of the composite.Template:R
A convex polyhedron in which all of its faces are regular is a Johnson solid, and the elongated square bipyramid is one of them, denoted as <math> J_{15} </math>, the fifteenth Johnson solid.Template:R
PropertiesEdit
Given that <math> a </math> is the edge length of an elongated square pyramid. The height of an elongated square pyramid can be calculated by adding the height of an equilateral square pyramid and a cube. The height of a cube is the same as the edge length of a cube's side, and the height of an equilateral square pyramid is <math> (1/\sqrt{2})a </math>. Therefore, the height of an elongated square bipyramid is:Template:R <math display="block"> a + \frac{1}{\sqrt{2}}a = \left(1 + \frac{\sqrt{2}}{2}\right)a \approx 1.707a. </math> Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:Template:R <math display="block"> \left(5 + \sqrt{3}\right)a^2 \approx 6.732a^2. </math> Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:Template:R <math display="block"> \left(1 + \frac{\sqrt{2}}{6}\right)a^3 \approx 1.236a^3. </math>
The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group <math> C_{4v} </math> of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:Template:R
- The dihedral angle of an elongated square bipyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces, <math> \arccos(-1/3) \approx 109.47^\circ </math>,
- The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those, <math> \pi/2 = 90^\circ </math>,
- The dihedral angle of an equilateral square pyramid between square and triangle is <math> \arctan \left(\sqrt{2}\right) \approx 54.74^\circ </math>. Therefore, the dihedral angle of an elongated square bipyramid between triangle-to-square, on the edge where the equilateral square pyramids attach the cube, is <math display="block"> \arctan\left(\sqrt{2}\right) + \frac{\pi}{2} \approx 144.74^\circ. </math>