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Composite Solids

Solids made up of two or more solids.

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Composite Solids

Composite Solids

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.

Finding the Volume of a Parallelogram-Based Prism

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*}

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

Finding the Surface Area

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*}” 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*}

Finding the Volume

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*}

Examples

Example 1

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

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*}

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

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

Example 2

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

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*}

Example 3

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

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*}

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*}

Review

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*}. 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*}.

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*} per minute, how long do you have to eat your treat before it all melts? Give your answer to the nearest minute.
1. For exercise 12, answer the following:
1. What is the surface area of a cylinder?
2. Adjust your answer from part a for the case where \begin{align*}r = h\end{align*}.
3. What is the surface area of a sphere?
4. What is the relationship between your answers to parts b and c? Can you explain this?

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

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Vocabulary Language: English

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.