- Source: Elongated square pyramid
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.
Construction
The elongated square bipyramid is a composite, since it can constructed by attaching two equilateral square pyramids onto the faces of a cube that are opposite each other, a process known as elongation. This construction involves the removal of those two squares and replacing them with those pyramids, resulting in eight equilateral triangles and four squares as their faces. 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
J
15
{\displaystyle J_{15}}
, the fifteenth Johnson solid.
Properties
Given that
a
{\displaystyle a}
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
(
1
/
2
)
a
{\displaystyle (1/{\sqrt {2}})a}
. Therefore, the height of an elongated square bipyramid is:
a
+
1
2
a
=
(
1
+
2
2
)
a
≈
1.707
a
.
{\displaystyle a+{\frac {1}{\sqrt {2}}}a=\left(1+{\frac {\sqrt {2}}{2}}\right)a\approx 1.707a.}
Its surface area can be calculated by adding all the area of four equilateral triangles and four squares:
(
5
+
3
)
a
2
≈
6.732
a
2
.
{\displaystyle \left(5+{\sqrt {3}}\right)a^{2}\approx 6.732a^{2}.}
Its volume is obtained by slicing it into an equilateral square pyramid and a cube, and then adding them:
(
1
+
2
6
)
a
3
≈
1.236
a
3
.
{\displaystyle \left(1+{\frac {\sqrt {2}}{6}}\right)a^{3}\approx 1.236a^{3}.}
The elongated square pyramid has the same three-dimensional symmetry group as the equilateral square pyramid, the cyclic group
C
4
v
{\displaystyle C_{4v}}
of order eight. Its dihedral angle can be obtained by adding the angle of an equilateral square pyramid and a cube:
The dihedral angle of an elongated square bipyramid between two adjacent triangles is the dihedral angle of an equilateral triangle between its lateral faces,
arccos
(
−
1
/
3
)
≈
109.47
∘
{\displaystyle \arccos(-1/3)\approx 109.47^{\circ }}
,
The dihedral angle of an elongated square bipyramid between two adjacent squares is the dihedral angle of a cube between those,
π
/
2
=
90
∘
{\displaystyle \pi /2=90^{\circ }}
,
The dihedral angle of an equilateral square pyramid between square and triangle is
arctan
(
2
)
≈
54.74
∘
{\displaystyle \arctan \left({\sqrt {2}}\right)\approx 54.74^{\circ }}
. 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
arctan
(
2
)
+
π
2
≈
144.74
∘
.
{\displaystyle \arctan \left({\sqrt {2}}\right)+{\frac {\pi }{2}}\approx 144.74^{\circ }.}
See also
Elongated square bipyramid
References
External links
Weisstein, Eric W., "Johnson solid" ("Elongated square pyramid") at MathWorld.
Kata Kunci Pencarian:
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- Elongated square bipyramid
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- Johnson solid
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- J8
