Solid geometry

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A solid shape, or solid, is any portion of space bounded by one or more surfaces, plane or curved. These surfaces are called the faces of the solid, and the intersections of adjacent faces are called edges.


Euler's formula

The number of faces (F), vertices(V), and edges(E) of a solid bound by plane faces are related by the formula F + V = E + 2. In the case of a cuboid, F= 6, V =8, E =12.



A parallelopiped is a solid bounded by three pairs of parallel plane faces.

  • Each of the six faces of a parallelepiped is a parallelogram.
  • Opposite faces are congruent.
  • The four diagonals of a parallelopiped are concurrent and bisect one another.



A parallelopiped whose faces are rectangular is called a cuboid. The three dimensions associated with a cuboid are its length, breadth and height (l , b, and h).



  • The length of the three pairs of face diagonals are


  • The length of the four equal body diagonals


  • The total surface area of the cuboid = 2 (lb + bh + hl)
  • Volume of cuboid = lbh



A cube is a parallelopiped all of whose faces are squares.

  • Total surface area of cube = 6a2
  • Volume of the cube = a3
  • Length of the face diagonal = √2 a
  • Length of the body diagonal = √3 a
  • Radius of the circumscribed sphere =  (√3 a)/2
  • Radius of the inscribed sphere = a/2
  • Radius of the sphere tangent to edges = a/√2



A pyramid is a solid bounded by plane faces, of which one, called the base, is any rectilinear figure, and the rest are triangles having a common vertex at some point, not in the plane of the base. The slant height of a pyramid is the height of its triangular faces. The height of a pyramid is the length of the perpendicular dropped from the vertex to the base.



In a pyramid with n sided regular polygon as its base,

  • Total number of vertices = n+1
  • Curved surface area of the pyramid


  • Total surface area of the pyramid


  • Volume




A tetrahedron is a pyramid which has four congruent equilateral triangles as its four faces.

  • Total number of vertices = 4
  • The four lines which join the vertices of a tetrahedron to the centroids of the opposite faces meet at a point which divides them in the ratio 3:1.
  • Total surface area of a tetrahedron = (√3 a2)/4  x 4 = √3 a2
  • Height of a tetrahedron = (√6 a)/3
  • Volume of a tetrahedron =  (√2 a3)/12



A prism is a solid bounded by plane faces, of which two, called the ends, are congruent figures in parallel planes and the others, called side faces are parallelograms. The ends of a prism may be triangles, quadrilaterals, or polygons of any number of sides.


A triangular, hexagonal, pentagonal and a right-circular prism


  • The side edges of every prism are all parallel and equal.
  • A prism is said to be right if the side edges are perpendicular to the ends: In this case, the side faces are rectangles.
  • The surface area of a prism = perimeter of the base x height + 2 x area of the base.
  • Volume of the right prism = Area of the base x height


Right circular cylinder

A right circular cylinder is a right prism whose base is a circle.

  • Curved surface area = 2πrh
  • Total surface area = 2πrh + πr2
  • Volume =  πr2h


Right circular cone

A right circular cone is a pyramid whose base is a circle.

  • Slant height


  • Curved surface area of the cone =  πrl
  • Total surface area of the cone =  πrl + πr2
  • Volume of the cone



Cone and frustum


Frustum of a cone

When a right circular cone is cut by a plane parallel to the base, the remaining portion is known as the frustum.

  • Slant height


  • Curved surface area of the cone = π(r + R)l
  • Total surface area of the cone =  π(r+R)l + π(r2 + R2)
  • Volume of the cone = π(r+R)2h /3



A sphere is a set of all points in space which are at a fixed distance from a given point. The fixed point is called the centre of the sphere, and the fixed distance is the radius of the sphere.


Sphere and shell


  • Surface area of the sphere =  4πr2
  • Volume of the sphere 



Spherical shell

For a hollow shell with inner and outer radii of r and R, respectively.

  • Volume of the shell