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Volume of Regularly-shaped Solids

according to oxford dictionary,solid means hard or firm.

A regularly shaped solid is an object with a definite shape that can always be described. Each regularly shaped solid have a known geometrical shape and hence can be identified by name.

Some of the common known regular solids includes:

Cube

A cube is a six sided object with all its edges equal in length. A cube has a solid shape with six square faces all equal in area and lengths.

The cube

The volume of a cube (Vcube) is given by Vcube= l x l x l = l3 where l is the length of the edge of the cube.

cuboid

A cuboid is an object with six faces where each pair of the opposite faces are equal in shape and size. Cuboid means “like a cube” because it has the same shape with a cube, except that all its sides are not equal.

The figure below shows a cuboid with one edge named length, another one named width and the other one named height.

Volume of a cuboid (Vcuboid) will be given by Vcuboid = Length x Width x Height

Cylinder

A cylinder is a three dimensional object consisting of two parallel circular surfaces that are connected by a curved surface. The distance between the two circular faces is a fixed distance and is usually refereed to as the height of the cylinder. There is an imaginary line that passes through the center of the circles and perpendicular to the circles known as the axis.

A cylinder with radius r and height h
A cylinder with radius r and height h

Volume of a cylinder (Vcylinder) is given by Vcylinder = BaseArea(BA) x height(h) where BaseArea is the area of one of the circular face given by Area (A) = πr2 hence Vcylinder = πr2 h

Sphere

A sphere is a geometrical object that is round in shape and is defined in a three-dimensional space without any face.

showing a sphere with radius r

Volume of a sphere (Vsphere) will be given by ;

Vsphere = (4/3)πr3

where r the radius and π a mathematical constant.

Cone

A cone is a three-dimensional shape with a flat circular base and a curved surface that forms a sharp point at the top. The sharp point is called the vertex.

The three parts that makes a cone are its radius, height, and slanting height. Radius r is the distance between the center of the circular base to any point on the circumference of the base.

The slant-height l is defined as the distance between the vertex of the cone to any point on the circumference of the circular base.

The height h of a cone is the distance between the vertex and the center of the circular base.

Figure below illustrates a cone

Volume of a cone (Vcone) will be given by Vcone = (1/3) πr2h.

but πr2h = volume of a cylinder.

hence Vcone = (1/) x Volume of a cylinder

Prisms

An octagonal prism
An octagonal prism

A prism is a three-dimensional object with two identical surfaces facing each other usually referred to as the bases of a prism. The base of the prism is usually called the cross-sectional area.

Length of the prism is distance between the two identical surfaces.

The base of the prism can assume varied shapes hence we have different types of prisms like:

  • square prism
  • triangular prism
  • rectangular prism
  • pentagonal prism
  • hexagonal prism
  • octagonal prism
  • nonagonal prism
  • decagonal prism
  • hendecagonal prism
  • Dodecagonal prism
  • tridecagonal/triskaidecagonal prism
  • tetradecagonal prism
  • pentadecagonal prism
  • e.t.c.

To get the volume of the prism, you simply gets area of the base and multiply it with the length of the prism. hence

volume a prism = cross-section Area(A) x length (l).

Volume of a Hexagonal prism

The hexagonal prism is a prism with hexagonal base. The word hexagonal comes from the word hexagon. In geometry, a hexagon is a six-sided polygon.

so volume of Hexagonal prism is given as a product of the area of the hexagonal base and the length between the two hexagonal ends.

A regular hexagon has six sides each with the same length. By drawing lines from vertices that are joining at the center of the hexagon, six isosceles triangles can be obtained from the hexagon. The area of the hexagon is equal to area of one triangle multiplied by number of triangles.

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