Surface Area and Volume Formulas

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Find the Volume and Surface Area of 3D Polygons

Finding the volume of three-dimensional polygons is easy if you know the formula to use.
Finding the volume of three-dimensional polygons is easy if you know the formula to use. Rafe Swan / Getty Images

Surface area and volume formulas are part of the math used in common science calculations. You may need to calculate surface area and volume to determine density, pressure and concentration.

Here is a list of surface area and volume formulas to use as a handy reference, for shapes including spheres, cones, cubes, prisms, cylinders, and pyramids.

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Sphere Surface Area and Volume Formulas

Sphere
Sphere. Todd Helmenstine

If you spin a circle into three dimensions, a sphere is formed. A sphere is a shape where the distance from the center to the edge is the same in all directions. This distance is called the radius ( r ).

Surface area = 4πr2
Volume = 4/3πr3

03
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Cone Surface Area and Volume Formulas

Cone
Cone. Todd Helmenstine

A cone is a pyramid with a circular base of radius ( r ) and the side length ( s ) is the length of the side.

s = √( r2 + h2 )

The surface area is the sum of the area of the base and area of the side.
Area of base: πr2
Area of side: πrs

Surface Area = πr2 + πrs
Volume = 1/3( πr2h )

04
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Rectangular Solid Surface Area and Volume Formulas

Box or Rectangular Solid
Box or Rectangular Solid. Todd Helmenstine

Surface Area = 2(lh) + 2(lw) + 2(wh)
Volume = l x h x w


where
l = length
h = height
w = width

See an example for calculating cubic feet of assorted shapes.

05
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Prism Surface Area and Volume Formulas

Prism
Prism. Todd Helmenstine

A prism can be described as a stack of shapes. The figure shows a prism of triangles stacked d thick, but any shape could be used.


Surface area = 2A + Pd


whereA = area of the base shapeP = perimeter of base shaped = height of prism

Volume = Ad

06
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Cylinder Surface Area and Volume Formulas

Cylinder
Cylinder. Todd Helmenstine

A cylinder is a prism with a circular base.


Surface Area = 2πr2 + 2πrh
Volume = πr2h

07
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Pyramid Surface Area and Volume Formulas

Pyramid
Pyramid. Todd Helmenstine

A pyramid is a solid figure with a polygonal base and triangular faces that meet at a common point over the center of the base.
The height ( h ) is the distance from the base to the apex or top of the pyramid.
The side length ( s ) is the height of the face triangles.
The perimeter ( P ) and the area ( A ) of the base is calculated according to the shape of the base.

Surface Area = ( ½ x P x s ) + A
Volume = 1/3 Ah


The figure shows a pyramid with a square base ( a = b ) with equilateral triangles for faces.

Surface area = a2 + √3( a2 )
Volume = √5(a3/6)