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Composite Solids

Solids made up of two or more solids.

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Composite Solids

What if you built a solid three-dimensional house model consisting of a pyramid on top of a square prism? How could you determine how much two-dimensional and three-dimensional space that model occupies? After completing this Concept, you'll be able to find the surface area and volume of composite solids like this one.

Watch This

CK-12 Foundation: Chapter11CompositeSolidsA

Learn more about the surface area of joined solids by watching the video at this link.

Guidance

A composite solid is a solid that is composed, or made up of, two or more solids. The solids that it is made up of are generally prisms, pyramids, cones, cylinders, and spheres. In order to find the surface area and volume of a composite solid, you need to know how to find the surface area and volume of prisms, pyramids, cones, cylinders, and spheres. For more information on any of those specific solids, consult the Concept that focuses on them. This Concept will assume knowledge of those five solids.

Example A

Find the volume of the solid below.

This solid is a parallelogram-based prism with a cylinder cut out of the middle.

\begin{align*}V_{prism} &= (25 \cdot 25)30=18,750 \ cm^3\\ V_{cylinder} &= \pi (4)^2 (30)=480 \pi \ cm^3\end{align*}VprismVcylinder=(2525)30=18,750 cm3=π(4)2(30)=480π cm3

The total volume is \begin{align*}18750 - 480 \pi \approx 17,242.04 \ cm^3\end{align*}18750480π17,242.04 cm3.

Example B

Find the surface area of the following solid.

This solid is a cylinder with a hemisphere on top. Because it is one fluid solid, we would not include the bottom of the hemisphere or the top of the cylinder because they are no longer on the surface of the solid. Below, “\begin{align*}LA\end{align*}LA” stands for lateral area.

\begin{align*}SA&=LA_{cylinder}+LA_{hemisphere}+A_{base \ circle}\\ & = \pi rh+ \frac{1}{2} 4 \pi r^2+ \pi r^2\\ & = \pi (6)(13)+2 \pi 6^2+ \pi 6^2\\ & = 78 \pi +72 \pi +36 \pi \\ & = 186 \pi \ in^2\end{align*}SA=LAcylinder+LAhemisphere+Abase circle=πrh+124πr2+πr2=π(6)(13)+2π62+π62=78π+72π+36π=186π in2

Example C

Find the volume of the following solid.

To find the volume of this solid, we need the volume of a cylinder and the volume of the hemisphere.

\begin{align*}V_{cylinder} & = \pi 6^2 (13)=78 \pi \\ V_{hemisphere}& = \frac{1}{2} \left ( \frac{4}{3} \pi 6^3 \right )=36 \pi\\ V_{total} & = 78 \pi +36 \pi =114 \pi \ in^3\end{align*}VcylinderVhemisphereVtotal=π62(13)=78π=12(43π63)=36π=78π+36π=114π in3

Watch this video for help with the Examples above.

CK-12 Foundation: Chapter11CompositeSolidsB

Vocabulary

A composite solid is a solid that is composed, or made up of, two or more solids. Surface area is a two-dimensional measurement that is the total area of all surfaces that bound a solid. Volume is a three-dimensional measurement that is a measure of how much three-dimensional space a solid occupies.

Guided Practice

1. Find the volume of the composite solid. All bases are squares.

2. Find the volume of the base prism. Round your answer to the nearest hundredth.

3. Using your work from #2, find the volume of the pyramid and then of the entire solid.

Answers:

1. This is a square prism with a square pyramid on top. Find the volume of each separeatly and then add them together to find the total volume. First, we need to find the height of the pyramid portion. The slant height is 25 and the edge is 48. Using have of the edge, we have a right triangle and we can use the Pythagorean Theorem. \begin{align*}h=\sqrt{25^2-24^2}=7\end{align*}h=252242=7

\begin{align*}V_{prism}&=(48)(48)(18)=41472 \ cm^3\\ V_{pyramid}&=\frac{1}{3} (48^2)(7)=5376 \ cm^3\end{align*}VprismVpyramid=(48)(48)(18)=41472 cm3=13(482)(7)=5376 cm3

The total volume is \begin{align*}41472 + 5376 = 46,848 \ cm^3\end{align*}41472+5376=46,848 cm3.

2. Use what you know about prisms.

\begin{align*}V_{prism}&=B \cdot h \\ V_{prism}&=(4\cdot 4)\cdot 5\\ V_{prism}&=80in^3\end{align*}VprismVprismVprism=Bh=(44)5=80in3

3. Use what you know about pyramids.

\begin{align*}V_{pyramid}&=\frac{1}{3} B \cdot h \\ V_{pyramid}&=\frac{1}{3}(4 \cdot 4)(6)\\ V_{pyramid}&=32in^3\end{align*}VpyramidVpyramidVpyramid=13Bh=13(44)(6)=32in3

Now find the total volume by finding the sum of the volumes of each solid.

\begin{align*}V_{total}&=V_{prism}+V_{pyramid}\\ V_{total}&=112 in^3\end{align*}VtotalVtotal=Vprism+Vpyramid=112in3

Practice

Find the volume of the composite solids below. Round your answers to the nearest hundredth.

  1. The bases are squares. Find the volume of the green part.
  2. A cylinder fits tightly inside a rectangular prism with dimensions in the ratio 5:5:7 and volume \begin{align*}1400 \ in^3\end{align*}1400 in3. Find the volume of the space inside the prism that is not contained in the cylinder.

Find the surface area and volume of the following shapes. Leave your answers in terms of \begin{align*}\pi\end{align*}π.

  1. You may assume the bottom is open.
  2. A sphere has a radius of 5 cm. A right cylinder has the same radius and volume. Find the height and total surface area of the cylinder.

Tennis balls with a 3 inch diameter are sold in cans of three. The can is a cylinder. Assume the balls touch the can on the sides, top and bottom.

  1. What is the volume of one tennis ball?
  2. What is the volume of the space not occupied by the tennis balls? Round your answer to the nearest hundredth.

One hot day at a fair you buy yourself a snow cone. The height of the cone shaped container is 5 in and its radius is 2 in. The shaved ice is perfectly rounded on top forming a hemisphere.

  1. What is the volume of the ice in your frozen treat?
  2. If the ice melts at a rate of \begin{align*}2 \ in^3\end{align*}2 in3 per minute, how long do you have to eat your treat before it all melts? Give your answer to the nearest minute.

Find the volume of the composite solids. Round your answer to the nearest hundredth.

Vocabulary

Pythagorean Theorem

Pythagorean Theorem

The Pythagorean Theorem is a mathematical relationship between the sides of a right triangle, given by a^2 + b^2 = c^2, where a and b are legs of the triangle and c is the hypotenuse of the triangle.

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