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11.4: Volume of Prisms and Cylinders

Difficulty Level: At Grade Created by: CK-12

Learning Objectives

  • Find the volume of prisms and cylinders.

Review Queue

  1. Define volume in your own words.
  2. What is the surface area of a cube with 3 inch sides?
  3. A regular octahedron has 8 congruent equilateral triangles as the faces.
    1. If each edge is 4 cm, what is the slant height for one face?
    2. What is the surface area of one face?
    3. What is the total surface area?

Know What? Let’s fill the pool it with water. The shallow end is 4 ft. and the deep end is 8 ft. The pool is 10 ft. wide by 25 ft. long. How many cubic feet of water is needed to fill the pool?

Volume of a Rectangular Prism

Volume: The measure of how much space a three-dimensional figure occupies.

Another way to define volume would be how much a three-dimensional figure can hold. The basic unit of volume is the cubic unit: cubic centimeter \begin{align*}(cm^3)\end{align*}, cubic inch \begin{align*}(in^3)\end{align*}, cubic meter \begin{align*}(m^3)\end{align*}, cubic foot \begin{align*}(ft^3)\end{align*}.

Volume of a Cube Postulate: \begin{align*}V=s^3\end{align*}.

\begin{align*}V = s \cdot s \cdot s=s^3\end{align*}

What this postulate tells us is that every solid can be broken down into cubes. For example, if we wanted to find the volume of a cube with 9 inch sides, it would be \begin{align*}9^3=729 \ in^3\end{align*}.

Volume Congruence Postulate: If two solids are congruent, then their volumes are congruent.

These prisms are congruent, so their volumes are congruent.

Example 1: Find the volume of the right rectangular prism below.

Solution: Count the cubes. The bottom layer has 20 cubes, or \begin{align*}4 \times 5\end{align*}, and there are 3 layers. There are 60 cubes. The volume is also \begin{align*}60 \ units^3\end{align*}.

Each layer in Example 1 is the same as the area of the base and the number of layers is the same as the height. This is the formula for volume.

Volume of a Rectangular Prism: \begin{align*}V=l \cdot w \cdot h\end{align*}.

Example 2: A typical shoe box is 8 in by 14 in by 6 in. What is the volume of the box?

Solution: We can assume that a shoe box is a rectangular prism.

\begin{align*}V=(8)(14)(6)=672 \ in^2\end{align*}

Volume of any Prism

Notcie that \begin{align*}l \cdot w\end{align*} is equal to the area of the base of the prism, which we will re-label \begin{align*}B\end{align*}.

Volume of a Prism: \begin{align*}V=B \cdot h\end{align*}.

\begin{align*}B\end{align*}” is not always going to be the same. So, to find the volume of a prism, you would first find the area of the base and then multiply it by the height.

Example 3: You have a small, triangular prism shaped tent. How much volume does it have, once it is set up?

Solution: First, we need to find the area of the base.

\begin{align*}B &= \frac{1}{2} (3)(4)=6 \ ft^2.\\ V &= Bh=6(7)=42 \ ft^3\end{align*}

Even though the height in this problem does not look like a “height,” it is, according to the formula. Usually, the height of a prism is going to be the last length you need to use.

Oblique Prisms

Recall that oblique prisms are prisms that lean to one side and the height is outside the prism. What would be the volume of an oblique prism? Consider to piles of books below.

Both piles have 15 books, which means they will have the same volume. Cavalieri’s Principle says that leaning does not matter, the volumes are the same.

Cavalieri’s Principle: If two solids have the same height and the same cross-sectional area at every level, then they will have the same volume.

If an oblique prism and a right prism have the same base area and height, then they will have the same volume.

Example 4: Find the area of the oblique prism below.

Solution: This is an oblique right trapezoidal prism. Find the area of the trapezoid.

\begin{align*}B &= \frac{1}{2} (9)(8+4)=9(6)=54 \ cm^2\\ V &= 54(15)=810 \ cm^3\end{align*}

Volume of a Cylinder

If we use the formula for the volume of a prism, \begin{align*}V = Bh\end{align*}, we can find the volume of a cylinder. In the case of a cylinder, the base is the area of a circle. Like a prism, Cavalieri’s Principle holds.

Volume of a Cylinder: \begin{align*}V=\pi r^2 h\end{align*}.

Example 5: Find the volume of the cylinder.

Solution: If the diameter is 16, then the radius is 8.

\begin{align*}V=\pi 8^2 (21)=1344 \pi \ units^3\end{align*}

Example 6: Find the volume of the cylinder.

Solution: \begin{align*}V=\pi 6^2 (15)=540 \pi \ units^3\end{align*}

Example 7: If the volume of a cylinder is \begin{align*}484 \pi \ in^3\end{align*} and the height is 4 in, what is the radius?

Solution: Solve for \begin{align*}r\end{align*}.

\begin{align*}484 \pi &= \pi r^2 (4)\\ 121 &= r^2\\ 11 &= r\end{align*}

Example 8: Find the volume of the solid below.

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

\begin{align*}V_{prism} &= (25 \cdot 25)30=18750 \ cm^3\\ V_{cylinder} &= \pi (4)^2 (30)=480 \pi \ cm^3\end{align*}

The total volume is \begin{align*}18750 - 480 \pi \approx 17242.04 \ cm^3\end{align*}.

Know What? Revisited Even though it doesn’t look like it, the trapezoid is the base of this prism. The area of the trapezoids are \begin{align*}\frac{1}{2} (4+8)25=150 \ ft^2\end{align*}. \begin{align*}V=150(10)=1500 \ ft^3\end{align*}

Review Questions

  • Question 1 uses the volume formula for a cylinder.
  • Questions 2-4 are similar to Example 1.
  • Questions 5-18 are similar to Examples 2-6.
  • Questions 19-24 are similar to Example 7.
  • Questions 25-30 are similar to Example 8.
  1. Two cylinders have the same surface area. Do they have the same volume? How do you know?
  2. How many one-inch cubes can fit into a box that is 8 inches wide, 10 inches long, and 12 inches tall? Is this the same as the volume of the box?
  3. A cereal box in 2 inches wide, 10 inches long and 14 inches tall. How much cereal does the box hold?
  4. A can of soda is 4 inches tall and has a diameter of 2 inches. How much soda does the can hold? Round your answer to the nearest hundredth.
  5. A cube holds \begin{align*}216 \ in^3\end{align*}. What is the length of each edge?
  6. A cube has sides that are 8 inches. What is the volume?
  7. A cylinder has \begin{align*}r = h\end{align*} and the radius is 4 cm. What is the volume?
  8. A cylinder has a volume of \begin{align*}486 \pi \ ft.^3\end{align*}. If the height is 6 ft., what is the diameter?

Use the right triangular prism to answer questions 9 and 10.

  1. What is the length of the third base edge?
  2. Find the volume of the prism.
  3. Fuzzy dice are cubes with 4 inch sides.
    1. What is the volume of one die?
    2. What is the volume of both dice?
  4. A right cylinder has a 7 cm radius and a height of 18 cm. Find the volume.

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

Algebra Connection Find the value of \begin{align*}x\end{align*}, given the surface area.

  1. \begin{align*}V=504 \ units^3\end{align*}
  2. \begin{align*}V=6144 \pi \ units^3\end{align*}
  3. \begin{align*}V=2688 \ units^3\end{align*}
  4. The area of the base of a cylinder is \begin{align*}49 \pi \ in^2\end{align*} and the height is 6 in. Find the volume.
  5. The circumference of the base of a cylinder is \begin{align*}80 \pi \ cm\end{align*} and the height is 15 cm. Find the volume.
  6. The lateral surface area of a cylinder is \begin{align*}30 \pi \ m^2\end{align*} and the circumference is \begin{align*}10 \pi \ m\end{align*}. What is the volume of the cylinder?

The bases of the prism are squares and a cylinder is cut out of the center.

  1. Find the volume of the prism.
  2. Find the volume of the cylinder in the center.
  3. Find the volume of the figure.

This is a prism with half a cylinder on the top.

  1. Find the volume of the prism.
  2. Find the volume of the half-cylinder.
  3. Find the volume of the entire figure.

Review Queue Answers

  1. The amount a three-dimensional figure can hold.
  2. \begin{align*}54 \ in^2\end{align*}
    1. \begin{align*}2 \sqrt{3}\end{align*}
    2. \begin{align*}\frac{1}{2} \cdot 4 \cdot 2 \sqrt{3} = 4 \sqrt{3}\end{align*}
    3. \begin{align*}8 \cdot 4 \sqrt{3} = 32 \sqrt{3}\end{align*}

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