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Prism, Pyramid, Cylinder, Cone,Sphere and other Solid Geometric Shapes – Definitions, Area and Volume formulas

Posted in Math Theory by Lia on 04/03/2013

General_3d_figures_2

Below is a table with definitions, area and volume formulas, for the body shapes shown on the right hand side.

 

 

 

 

For a table with perimeters and areas for the most common 2D shapes, click  here.

 

 

Name/Description

Visual

representation

Lateral

Surface Area

 ({}{A_l}{})

Total Surface Area ({}{A_t}{})

Volume (V)

Polyhedron

A geometric solid that is bounded by plane polygons.

 

 Prism

A polyhedron with two congruent parallel n-sided polygons (bases) and n parallelograms as sides between the two basis.

 

Right Prism

A prism whose sides’ joining edges and sides’ faces are perpendicular to the bases.

Its sides are rectangles.

 Prism_2

tabular{11}{11}{{{A_l} = h* {p}}}

 OR {}{A_l}{} can be calculated as

tabular{11}{10}{{{A_l} = {sum}{}{of}}}

tabular{11}{01}{{{side}{}{areas}}}

where the sides are rectangles (look here for rectangle formula).

 tabular{11}{11}{{{A_t} = 2{A_b} + {A_l}}}

To calculate the {}{A_b}{}, figure out what is the base’s shape and look here for its formula.

tabular{11}{11}{{V = h*{A_b}}}

To calculate the {}{A_b}{}, figure out what is the base’s shape and look here for its formula.

 

Oblique Prism

A prism whose sides’ joining edges and sides’ faces are not perpendicular to the bases.

Its sides are parallelograms.

Oblique_prism_1

tabular{11}{11}{{{A_l} = h* {p}}}

OR {}{A_l}{} can be calculated as

tabular{11}{10}{{{A_l} = {sum}{}{of}}}

tabular{11}{01}{{{side}{}{areas}}}

where the sides are parallelograms (look here for parallelogram formula).

 

 tabular{11}{11}{{{A_t} = 2{A_b} + {A_l}}}

To calculate the {}{A_b}{}, figure out what is the base’s shape and look here for its formula.

 

tabular{11}{11}{{V = h*{A_b}}}

To calculate the {}{A_b}{}, figure out what is the base’s shape and look here for its formula.

Parallelepiped

A prism whose bases are parallelograms.

In this case is a

Rectangular Parallelepiped

 A right prism whose bases are rectangles.

 Prallelipiped_5

tabular{11}{11}{{{A_l} = 2h(a + b)}}

 

 tabular{11}{10}{{{A_t} = 2ab}}

tabular{11}{01}{{+ 2h(a+b)}}

tabular{11}{11}{{V = h*a*b}}

Cube

A rectangular parallelepiped whose sides and  bases are equal squares.

Cube

 

tabular{11}{11}{{{A_l} = 4a^2}}

tabular{11}{11}{{{A_t} = 6{a^2}}}

 

 tabular{11}{11}{{V = {a^3}}}

Pyramid

A polyhedron with a polygonal base and triangles for sides.

 

 Pyramid

 

tabular{11}{10}{{{A_l} = {sum}{}{of}}}

tabular{11}{01}{{{side}{}{areas}}}

in our example:  

tabular{11}{10}{{{A_l} = {A_s1} + {A_s2}}}

tabular{11}{00}{{ + {A_s3} + {A_s4} + }}

tabular{11}{01}{{{A_s5} + {A_s6}}}

 tabular{11}{11}{{{A_t} = {A_b} + {A_l}}}

To calculate the {}{A_b}{}, figure out what is the base’s shape and look here for its formula.

tabular{11}{11}{{V = {1/3}( h*{A_b})}}

To calculate the {}{A_b}{}, figure out what is the base’s shape and look here for its formula.

Truncated Pyramid (Pyramidal Frustum)

A polyhedron with two polygonal parallel bases and trapezoids for sides.

OR

A portion of the pyramid that lies in between two parallel planes.

Its sides are trapezoids.

(for trapezoid formula look here)

 Truncated_pyramid

tabular{11}{10}{{{A_l} = {sum}{}{of}}}

tabular{11}{01}{{{side}{}{areas}}}

in our example:  

tabular{11}{10}{{{A_l} = {A_s1} + {A_s2}}}

tabular{11}{00}{{ + {A_s3} + {A_s4} + }}

tabular{11}{01}{{{A_s5} + {A_s6}}}

 

 tabular{11}{10}{{{A_t} = {A_b1} + }}

tabular{11}{01}{{{A_b2} + {A_l}}}

To calculate the {}{A_b1}{} and{}{A_b2}{}, figure out what are the bases’ shapes and look here for their formulas.

tabular{11}{10}{{V = }}

tabular{11}{00}{{{h/3}{[}{A_b1} + {A_b2} + }}

tabular{11}{01}{{ {sqrt{{A_b1}{A_b2}}}{]}}} 

 

To calculate the {}{A_b1}{} and{}{A_b2}{}, figure out what are the bases’ shapes and look here for their formulas.

 Cylinder

A geometric solid that has two parallel plane circular bases, connected by a curved surface.

 Right Cylinder

A cylinder that has  circular bases connected by a  curved surface perpendicular on the bases.

 In this case the base is a circle.

 Cylinder_1

 

tabular{11}{11}{{{A_l} = h* {p}}}

in our example:

tabular{11}{11}{{{A_l} = 2{pi}Rh}}

 

 tabular{11}{11}{{{A_t} = 2{A_b} + {A_l}}}

in our example:

tabular{11}{11}{{A_t = 2{pi}R(R + h)}}

tabular{11}{11}{{V = h*{A_b}}}

in our example:

tabular{11}{11}{{V = {{pi}{R^2}}h}}

 Oblique Cylinder

A cylinder that has  circular bases connected by a  curved surface that is not perpendicular on the bases.

In this case the base is a circle.

 Oblique_cylinder_2

 

tabular{11}{11}{{{A_l} = h* {p}}}

in our example:

tabular{11}{11}{{{A_l} = 2{pi}Rh}}

.

tabular{11}{11}{{{A_t} = 2{A_b} + {A_l}}}

in our example:

tabular{11}{11}{{A_t = 2{pi}R(R + h)}}

tabular{11}{11}{{V = h*{A_b}}}

in our example:

tabular{11}{11}{{V = {{pi}{R^2}}h}}

Cone

A geometric solid that  is bounded by a plane circular base and a surface generated by a straight line, the  generatrix or the slant height, passing through a fixed point, the vertex, and moving along a closed curve, the diretrix (the base’s perimeter).

Right Circular Cone

A cone that has  the axis passing through the vertex and center of the base,   perpendicular on the base.

The base is a circle. 

 Cone_2

tabular{11}{11}{{{A_l} = {pi}RS}}

where:

{S} = {sqrt{{R^2} + {h^2}}}

tabular{11}{11}{{{A_t} = {pi}R(R +S)}}

tabular{11}{11}{{V = {1/3} {pi}{R^2}h}}

Truncated Cone (Conical Frustum)

A portion of the cone that lies in between two parallel planes

 Truncated_cone_6

tabular{11}{11}{{{A_l} = {pi}S({R_1} + {R_2})}}

where:

{S} = {sqrt{{({R_1} - {R_2})^2} + {h^2}}}

        tabular{11}{10}{{{A_t} = {pi}{[}S({R_1} +{R_2})}}

       tabular{11}{01}{{ + {{R_1}^2} +  {{R_2}^2} {]}}}

tabular{11}{10}{{V = {{h{pi}}/3}{[}{{R_1}^2} + }}

tabular{11}{01}{{{{R_2}^2} + {{R_1}{R_2}}{]}}}

Sphere

A geometric solid that  is generated by a semicircle that rotates about an axis.

OR

A geometric solid that  is bounded by a surface generated by a set of points equidistant from a fixed point named center.

Sphere

 Sphere_2

tabular{11}{11}{{{A_l} = {4}{pi}{R^2}}}

 

tabular{11}{11}{{{A_t} = {A_l} = {4}{pi}{R^2}}}

 

tabular{11}{11}{{V = {4/3} {pi}{R^3}}}

 Spherical segment

A portion of the sphere that lies in between two parallel planes

 Spherical segment_5

tabular{11}{11}{{{A_l} = {2}{pi}Rh}}

tabular{11}{10}{{{A_t} = {A_l} + {2}{A_b} =}}

tabular{11}{01}{{{pi}({2}{R^2} + {{r_1}^2} + {{r_2}^2})}}

 

tabular{11}{10}{{V = }}

tabular{11}{00}{{{{{pi}h}/2} ({{r_1}^2} + {{r_2}^2})}}

tabular{11}{01}{{+ {{pi{h^3}}/6}}}

Spherical cap

Is a portion of a sphere that is above or below a plane that intersects that sphere.  

 Spherical cap_3

 

tabular{11}{11}{{{A_l} = {2}{pi}Rh}}

tabular{11}{10}{{{A_t} = {A_l} + {A_b} =}}

tabular{11}{01}{{{pi}({2}{R^2} + {{r}^2})}}

tabular{11}{10}{{V = {{{pi}{r^2}h}/2}}}

tabular{11}{01}{{+ {{pi{h^3}}/6}}}

Spherical sector

A geometric solid that  is generated by a circle sector  that rotates about an axis.

 Spherical sector_4

tabular{11}{11}{{{A_l} = {2}{pi}R({2}h + r)}}

tabular{11}{10}{{{A_t} = {A_l} = }}

tabular{11}{01}{{{2}{pi}R({2}h + r)}}

tabular{11}{11}{{V = {2/3} {pi}{R^2}h}}

 

 

 

 

 

 

 

 

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