Explainer: Volumes of Composite Solids

In this explainer, we will learn how to find volumes of composite solids consisting of two or more regular solids.

Definition: Composite Solid

A composite solid is a solid made of two or more well-defined solid objects.

In order to find the volume of a composite solid, one needs to identify the different parts it is made of. This decomposition allows working out the volume of each part independently. The volume of the composite solid is simply the sum of the volumes of its parts. For composite prisms, where the bases are a composite shape, the area of the bases is the sum of the areas of the parts it is made of. The volume of the composite prism is then given by multiplying the whole base area by the height of the prism.

Let’s work out some examples.

Example 1: Finding the Volume of a Composite Prism

Anthony made a cardboard house at school. The lower part of the house is a rectangular prism and the upper part is a triangular prism. Find the volume of the house.

Answer

This cardboard house is a composite prism: it has two parallel congruent shapes (the bases) and its cross-section is constant. The shape of its base is composite: it is made of a rectangle and a triangle.

To find the volume of this cardboard house, we need to find the areas of the rectangle 𝐴rectangle=lengthΓ—width and the triangle 𝐴triangle=baseΓ—height2: 𝐴rectangle=45Γ—20=900cm2,𝐴triangle=45Γ—182=405cm2,𝐴base=900+405=1,305cm2.

The volume of this composite prism is 𝑉prism=𝐴baseΓ—height=1,305Γ—17=22,185cm3.

Example 2: Finding the Volume of a Holed Cylinder

A roll of paper towels has the given dimensions. Determine, to the nearest hundredth, the volume of the roll.

Answer

Here, we need to find the volume of the roll, which is a cylinder with a cylindrical hole in the middle. The volume of the roll is given by the volume of the cylinder minus the volume of the hole.

The volume of a cylinder of radius π‘Ÿ and height β„Ž is 𝑉cylinder=𝐴baseΓ—β„Ž=πœ‹π‘Ÿ2Γ—β„Ž.

The volume of the big cylinder is then 𝑉bigcylinder=πœ‹Γ—ο€Ό1622Γ—30𝑉bigcylinder=πœ‹Γ—ο€Ό1622Γ—30=1,920πœ‹, and the volume of the hole is 𝑉hole=πœ‹π‘Ÿ2holeΓ—β„Ž=πœ‹Γ—ο€Ό422Γ—30𝑉hole=120πœ‹,𝑉roll=𝑉bigcylinderβˆ’π‘‰hole𝑉roll=1,920πœ‹βˆ’120πœ‹β‰…5,654.87cm3.

Example 3: Finding the Volume of a Composite Prism

The figure shows the design of a swimming pool. Work out, in cubic meters, the volume of water needed to fill the swimming pool completely.

Answer

The swimming pool is a composite prism made of a trapezoidal prism and a cuboid. The diagram shows how the shape of the bases is made of a trapezoid and a rectangle.

To find the volume of this composite prism, we need to find the areas of the rectangle 𝐴rectangle=lengthΓ—width and of the trapezoid 𝐴trapezoid=sumofparallelsides2Γ—height making up its base: 𝐴rectangle=30Γ—1.5=45m2,𝐴trapezoid=(20+15)2Γ—2.5=43.75m2,𝐴base=𝐴rectangle+𝐴trapezoid=45+43.75=88.75m2.

The volume of the prism is given by the area of its base multiplied by its height: 𝑉prism=𝐴baseΓ—height=88.75Γ—15=1,331.25m3.

Example 4: Finding the Volume of the Model of a Tree

By modeling the trunk of the tree as a cylinder and the head of the tree as a sphere, ignoring any air between the leaves and branches, work out an estimate for the volume of the tree seen in the given figure. Give your answer in terms of πœ‹.

Answer

Let us work out first the volume of the tree. It is modeled as a cylinder of diameter 1.5 ft and height 8 ft. The volume of a cylinder is given by 𝑉cylinder=πœ‹π‘Ÿ2β‹…β„Ž.

The radius is half the diameter; therefore, here π‘Ÿ=0.75ft. Substituting, we find 𝑉cylinder=πœ‹β‹…0.752β‹…8=4.5πœ‹ft3.

The branches and leaves are modeled as a sphere of diameter 9 ft. The volume of a sphere is given by 𝑉sphere=43πœ‹π‘Ÿ3.

The radius is half the diameter; therefore, here π‘Ÿ=4.5ft. Substituting, we find 𝑉sphere=43πœ‹β‹…4.53=121.5πœ‹ft3.

Adding the volumes of the tree and the branches and leaves, we find 𝑉sphere=126πœ‹ft3.

Example 5: Finding the Volume of a Composite Solid

The shape in the given figure consists of a cylinder with a hemisphere attached to each end. Work out its volume, giving your answer to two decimal places.

Answer

Let us start with the volume of the cylinder.

The volume of a cylinder is given by 𝑉cylinder=πœ‹π‘Ÿ2β‹…β„Ž.

Substituting, we find 𝑉cylinder=πœ‹32β‹…10β‰…282.743ft3.

A hemisphere of radius 3 feet is attached to each end of the cylinder, which means that we have two congruent hemispheres whose total volume is the volume of a whole sphere of radius 3 feet.

The volume of a sphere is given by 𝑉sphere=43πœ‹π‘Ÿ3.

Substituting, we find 𝑉sphere=43πœ‹33β‰…113.097ft3.

Adding both volumes together and rounding to two decimal places, we find 𝑉≅395.84ft3.

Key Points

  1. A composite solid is a solid made of two or more well-defined solid objects.
  2. To find the volume of a composite solid, we identify the different parts it is made of, work out the volume of each part independently, and sum up the volumes of its parts.
  3. For composite prisms, where the bases are a composite shape, the area of the bases is the sum of the areas of the parts it is made of. The volume of the composite prism is then given by multiplying the whole base area by the height of the prism.

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