Volume of irregular shapes

You can calculate the volume of irregular shapes easily once you know how to get the volume of a single three-dimensional shape.

The house below is made of two 3-d shapes: A triangular prism and a rectangular prism.

The roof of the house is the one shaped like a triangular prism.

Volume of house = volume of triangular prism + volume of rectangular prism.

volume of triangular prism = area of triangle × length of house.

volume of triangular prism =  
base of triangle × height of triangle / 2
  × length of house


length of house = 50 feet, base of triangle = 15 feet, and height of triangle = 10 feet

volume of triangular prism =  
15 × 10 / 2
  × 50


volume of triangular prism = 75 × 50 = 3750 feet3

volume of rectangular prism = 15 × 15 × 50 = 11250 feet3

Volume of house = 3750 + 11250 = 15000 feet3

Dimension of a house

A couple more great examples showing how to calculate the volume of irregular shapes


Ice cream cone:

When you order ice cream, you may have never realized that it could be a combination of half a sphere and a cone

Of course, we have to assume that the shape made by the ice cream scoop is half a sphere.

The shape of my ice cream shown below is half a sphere and then we put it on a cone. What is the volume?

Strawberry ice cream on a cone


It may be easier to see how to compute the volume if we remove the ice cream

Half a sphere on top of a cone
Pretend that the radius of the cone or r is 1.5 inches and the height of the cone or h is 6.5 inches here is how to compute the volume

Volume = volume of half the sphere + volume of the cone

Volume of a sphere =  
4 × π × r3 / 3


Volume of a sphere =  
42.39 / 3


Volume = 14.13 inches3

Since it is half a sphere, we get 7.065 inches3

Volume of a cone =  
π × r2 × h / 3


Volume of a cone =  
3.14 × 1.52 × 6.5 / 3


Volume of a cone =  
45.9225 / 3


Volume of cone = 15.3075 inches3

Volume of the whole thing is 7.065 + 15.3075 = 22.3725 inches3


A box containing 4 cream cones:

A box has 4 identical ice cream cones arranged inside of the box as shown below. The radius of the base of the cone is 1 inch and the height is 6 inches

Cones inside a box


What is the volume of the space left inside the box ?

Space left inside the box = volume of box - volume of 4 cones

Volume of box = 8 × 8 × 8 = 512 inches3

volume of a cone =  
π × r2 × h / 3


volume of a cone =  
3.14 × 12 × 6 / 3


volume of a cone = 6.28. Therefore, volume of 4 cones = 4 × 6.28 = 25.12

Volume left inside = 512 - 25.12 = 486.88 inches3


Hopefully you know now how to find the volume of irregular shapes!

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