text: Square Pyramid
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  Square Pyramid
  Square pyramid
  TypeJohnson
  J92 - J1 - J2
  Faces4 triangles
  1 square
  Edges8
  Vertices5
  Vertex configuration4(32.4)
  (34)
  Schläfli symbol( ) ? {4}
  Symmetry groupC4v, [4], (*44)
  Rotation groupC4, [4]+, (44)
  Dual polyhedronself
  Propertiesconvex
  Net
  3D model of a square pyramid
  
  In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has C4v symmetry. If all edges are equal, it is an equilateral square pyramid,[1] the Johnson solid J1.
  
  ## General square pyramid
  
  A possibly oblique square pyramid with base length l and perpendicular height h has volume:
  
  ${\displaystyle V={\frac {1}{3}}l^{2}h}$.
  
  ## Right square pyramid
  
  In a right square pyramid, all the lateral edges have the same length, and the sides other than the base are congruent isosceles triangles.
  
  A right square pyramid with base length l and height h has surface area and volume:
  
  ${\displaystyle A=l^{2}+l{\sqrt {l^{2}+(2h)^{2}}}}$
  ${\displaystyle V={\frac {1}{3}}l^{2}h}$.
  
  The lateral edge length is:
  
  ${\displaystyle {\sqrt {h^{2}+{{l^{2}} \over {2}}}}}$,
  
  and the slant height is:
  
  ${\displaystyle {\sqrt {h^{2}+{{l^{2}} \over {4}}}}}$.
  
  The dihedral angles are:
  
  between the base and a side: ${\displaystyle \arctan \left({{2\,h} \over {l}}\right)}$;
  between two sides: ${\displaystyle \arccos \left({{-l^{2}} \over {l^{2}+4\,h^{2}}}\right)}$.
  
  ## Equilateral square pyramid, Johnson solid J1
  
  If all the edges have the same length, then the sides are equilateral triangles, and the pyramid is an equilateral square pyramid, Johnson solid J1.
  
  The Johnson square pyramid can be characterized by a single edge length parameter l.
  
  The height h (from the midpoint of the square to the apex), the surface area A (including all five faces), and the volume V of an equilateral square pyramid are:
  
  ${\displaystyle h={\frac {1}{\sqrt {2}}}l}$
  ${\displaystyle A=(1+{\sqrt {3}})l^{2}}$
  ${\displaystyle V={\frac {\sqrt {2}}{6}}l^{3}.}$
  
  The dihedral angles of an equilateral square pyramid are:
  
  • Between the base and a side:
  ${\displaystyle \arctan {{\biggl (}{\sqrt {2}}{\biggl )}}\approx 54.73561^{\circ }.}$
  ${\displaystyle \arccos {\biggl (}{-1 \over 3}{\biggl )}\approx 109.47122^{\circ }.}$
  
  ## Graph
  
  A square pyramid can be represented by the Wheel graph W5.
  
  ## Related polyhedra and honeycombs
  
  Regular pyramids
  Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal Decagonal...
  Improper Regular Equilateral Isosceles
  A regular octahedron can be considered a square bipyramid, i.e. two Johnson square pyramids connected base-to-base. The tetrakis hexahedron can be constructed from a cube with short square pyramids added to each face. Square frustum is a square pyramid with the apex truncated.
  
  Square pyramids fill space with tetrahedra, truncated cubes or cuboctahedra.[2]
  
  ### Dual polyhedron
  
  The square pyramid is topologically a self-dual polyhedron. The dual edge lengths are different due to the polar reciprocation.
  
  Dual Square pyramid Net of dual
explanation: The document provides a detailed description of a square pyramid, including its geometric properties, formulas for volume and surface area, and related polyhedra. It does not contain a specific question or a question-and-answer pair.
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distilabel_metadata.content: Square Pyramid
  Get Square Pyramid essential facts below. View Videos or join the Square Pyramid discussion. Add Square Pyramid to your PopFlock.com topic list for future reference or share this resource on social media.
  Square Pyramid
  Square pyramid
  TypeJohnson
  J92 - J1 - J2
  Faces4 triangles
  1 square
  Edges8
  Vertices5
  Vertex configuration4(32.4)
  (34)
  Schläfli symbol( ) ? {4}
  Symmetry groupC4v, [4], (*44)
  Rotation groupC4, [4]+, (44)
  Dual polyhedronself
  Propertiesconvex
  Net
  3D model of a square pyramid
  
  In geometry, a square pyramid is a pyramid having a square base. If the apex is perpendicularly above the center of the square, it is a right square pyramid, and has C4v symmetry. If all edges are equal, it is an equilateral square pyramid,[1] the Johnson solid J1.
  
  ## General square pyramid
  
  A possibly oblique square pyramid with base length l and perpendicular height h has volume:
  
  ${\displaystyle V={\frac {1}{3}}l^{2}h}$.
  
  ## Right square pyramid
  
  In a right square pyramid, all the lateral edges have the same length, and the sides other than the base are congruent isosceles triangles.
  
  A right square pyramid with base length l and height h has surface area and volume:
  
  ${\displaystyle A=l^{2}+l{\sqrt {l^{2}+(2h)^{2}}}}$
  ${\displaystyle V={\frac {1}{3}}l^{2}h}$.
  
  The lateral edge length is:
  
  ${\displaystyle {\sqrt {h^{2}+{{l^{2}} \over {2}}}}}$,
  
  and the slant height is:
  
  ${\displaystyle {\sqrt {h^{2}+{{l^{2}} \over {4}}}}}$.
  
  The dihedral angles are:
  
  between the base and a side: ${\displaystyle \arctan \left({{2\,h} \over {l}}\right)}$;
  between two sides: ${\displaystyle \arccos \left({{-l^{2}} \over {l^{2}+4\,h^{2}}}\right)}$.
  
  ## Equilateral square pyramid, Johnson solid J1
  
  If all the edges have the same length, then the sides are equilateral triangles, and the pyramid is an equilateral square pyramid, Johnson solid J1.
  
  The Johnson square pyramid can be characterized by a single edge length parameter l.
  
  The height h (from the midpoint of the square to the apex), the surface area A (including all five faces), and the volume V of an equilateral square pyramid are:
  
  ${\displaystyle h={\frac {1}{\sqrt {2}}}l}$
  ${\displaystyle A=(1+{\sqrt {3}})l^{2}}$
  ${\displaystyle V={\frac {\sqrt {2}}{6}}l^{3}.}$
  
  The dihedral angles of an equilateral square pyramid are:
  
  • Between the base and a side:
  ${\displaystyle \arctan {{\biggl (}{\sqrt {2}}{\biggl )}}\approx 54.73561^{\circ }.}$
  ${\displaystyle \arccos {\biggl (}{-1 \over 3}{\biggl )}\approx 109.47122^{\circ }.}$
  
  ## Graph
  
  A square pyramid can be represented by the Wheel graph W5.
  
  ## Related polyhedra and honeycombs
  
  Regular pyramids
  Digonal Triangular Square Pentagonal Hexagonal Heptagonal Octagonal Enneagonal Decagonal...
  Improper Regular Equilateral Isosceles
  A regular octahedron can be considered a square bipyramid, i.e. two Johnson square pyramids connected base-to-base. The tetrakis hexahedron can be constructed from a cube with short square pyramids added to each face. Square frustum is a square pyramid with the apex truncated.
  
  Square pyramids fill space with tetrahedra, truncated cubes or cuboctahedra.[2]
  
  ### Dual polyhedron
  
  The square pyramid is topologically a self-dual polyhedron. The dual edge lengths are different due to the polar reciprocation.
  
  Dual Square pyramid Net of dual
distilabel_metadata.role: user
distilabel_metadata.raw_output_q_a_detector_0: The document provides a detailed description of a square pyramid, including its geometric properties, formulas for volume and surface area, and related polyhedra. It does not contain a specific question or a question-and-answer pair.
  
  Label: no-question