Volume of Prisms

Volume of Prisms exercise

• Identify the steps for calculating the volume of rectangular prisms.

  Hints

  All prisms have a 2-dimnesional shape on the bottom of the 3-dimensional figure. For example, a rectangular prism has a rectangular base.

  The area of the base is calculated by the area formula for the given figure.

  • For example, a rectangular prism has a rectangular base and the area of a rectangle is length $\cdot$ width, or $lw$, where $l$ represents length and $w$ represents width.

  The volume of a rectangular prism with length $15~\text{in}$, width $8~\text{in}$, and height $17~\text{in}$, can be calculated by the formula,

  • $V=15 ~\text{in} \cdot 8~\text{in}\cdot 17~\text{in}$
  • Therefore, the volume is $2,040~\text{in}^3$.

  Solution

  Volume is the amount of 3-dimnesional space an object occupies.

  When calculating the volume, you need to know two things:

  1. The area of the base.

  2. The height of the prism.

  Then, to get the volume, we multiply them together.

  • Volume $=$ area of the base $\cdot$ height

  The base of a rectangular prism is a rectangle and the area of a rectangle is the product of the two sides.

  • Area of Base $=$ length $\cdot$ width

  Therefore the volume of the rectangular prism is:

  • $V=$ length $\cdot$ width $\cdot$ height

  Let $l$ represent length, $w$ represent width, and $h$ represent height. The volume formula can be written as:

  • $V=lwh$

  Or more simply,

  • $V=Bh$ where $B$ represents the area of the base.

  The volume of a rectangular prism with length $47~\text{in}$, width $20~\text{in}$ and height $32~\text{in}$ can be calculated by the formula,

  • $V=$ $47~\text{in}\cdot 20~\text{in}\cdot 32~\text{in}$

  Therefore the volume is $30,080~$in$^3$.

• Sort the steps for calculating the volume of a triangular prism.

  Hints

  Remember to always start with the needed formula. Here that is $V=Bh$ where $B$ is the area of the prism's base, and $h$ is the height of the prism.

  Next, find the area of the base which is represented by $B$.

  $B$ can vary depending on the type of prism.

  • If the prism is rectangular, $B$ is the area of a rectangle, $lw$, where $l$ is the length of the rectangle and $w$ is the width of the rectangle.

  A triangular prism with height $18~$in and a triangular base that measures $16~$in across its base, and $12~$in high, is calculated using the following steps:

  1. $V=Bh$

  2. $B=\frac{1}{2}16~\text {in}\cdot 12~\text {in}$

  3 $B=96~\text{in}^2$

  4. $V=96~\text{in}^2\cdot 18~\text{in}$

  5. $V=1,728~\text{in}^3$

  Solution

  The following steps are used to calculate a triangular prism with height $17~\text{in}$, and a triangular base that measures $10~\text{in}$ across its base and $5~\text{in}$ high.

  1. $V=Bh$

  2. $B=\frac{1}{2}\cdot10~\text{in}\cdot 5~\text{in}$

  3. $B=25~\text{in}^2$

  4. $V=25~\text{in}^2\cdot 17~\text{in}$

  5. $V=425~\text{in}^3$

  Important Side Notes:

  1. $B$ represents the area of the base, and can vary depending on the type of prism.

  • If the prism is rectangular, $B$ is the area of a rectangle, $lw$, where $l$ is the length of the rectangle and $w$ is the width of the rectangle.
  • If the prism is traingular, $B$ is the area of a triangle, $\frac{1}{2}bh$, where $b$ is the base of the triangle and $h$ is the height of the triangle.

  2. There are two values for height in a triangular prism:

  • The height of the triangular base.

  • The height of the triangular prism.

  3. Always check that the units are the same. For example, this problem has units in inches but if one unit was feet, it would have to be converted to inches before calculating the volume.

• Calculate the volume of right prisms.

  Hints

  The volume of a prism is, $V=Bh$, where $B$ represents the area of the base and $h$ represents the height of the prism. The area of the base is different for each type of prism.

  The volume for the rectangular prism above is,

  • $V=lwh$
  • $V=15~\text{in}\cdot 3~\text{in}\cdot 13~\text{in}$
  • $V=585~\text{in}^3$

  The volume for the triangular prism above is,

  • $V=\frac{1}{2}\cdot b\cdot h_{1}\cdot h_{2}$
  • $V=\frac{1}{2}\cdot 27~\text{in}\cdot 11~\text{in}\cdot 42~\text{in}$
  • $V=6,237~\text{in}^3$

  Solution

  The volume of a prism is $V=Bh$, where $B$ represents the area of the base and $h$ represents the height of the prism.

  1. The rectangular prism with a length of $27$ inches, a width of $9$ inches, and a height $14$ inches has a volume of,

  • $V=lwh$
  • $V=27~\text{in}\cdot 9~\text{in}\cdot 14~\text{in}$
  • $V=3,402~\text{in}^3$

  2. The triangular prism with a height of $51$ inches, $42$ inch, and base height $19$ inches has a volume of,

  • $V=\frac{1}{2}\cdot b\cdot h_{1}\cdot h_{2}$
  • $V=\frac{1}{2}\cdot 42~\text{in}\cdot 19~\text{in}\cdot 51~\text{in}$
  • $V=20,349~\text{in}^3$

  3. The rectangular prism with a length of $24$ inches, a width of $7$ inches, and a height $11$ inches has a volume of,

  • $V=lwh$
  • $V=24~\text{in}\cdot 7~\text{in}\cdot 11~\text{in}$
  • $V=1,848~\text{in}^3$

  4. The cube with a side lengths of $13$ inches has a volume of,

  • $V=s\cdot s\cdot s$
  • $V=13~\text{in}\cdot 13~\text{in}\cdot 13~\text{in}$
  • $V=2,197~\text{in}^3$

  5. The triangular prism with a height of $37$ inches, $30$ inch, and base height $17$ inches has a volume of,

  • $V=\frac{1}{2}\cdot b\cdot h_{1}\cdot h_{2}$
  • $V=\frac{1}{2}\cdot 37~\text{in}\cdot 30~\text{in}\cdot 17~\text{in}$
  • $V=9,435~\text{in}^3$

• Solve real-world problems involving the volume of right prisms.

  Hints

  The volume of a prism is, $V=Bh$, where $B$ represents the area of the base and $h$ represents the height of the prism.

  • The area of a rectangular base is, $B=lw$.
  • The area of a triangular base is, $B=\frac{1}{2}bh$.

  The volume of a box that is $5$ feet long, $3$ feet wide and $7$ feet high is,

  • $V=lwh$
  • $V=5~\text{ft}\cdot 3~\text{ft}\cdot 7~\text{ft}$
  • $V=105~\text{ft}^3$

  The volume of a $6$ foot long tent with a triangular face that measures $4$ feet along the base and $5$ feet high is,

  • $V=\frac{1}{2}\cdot b\cdot h_{1}\cdot h_{2}$
  • $V=\frac{1}{2}\cdot 4~\text{ft}\cdot 5~\text{ft}\cdot 6~\text{in}$
  • $V=60~\text{ft}^3$

  Solution

  1. A family is moving from New York to California and is renting a truck to transport their boxes and furniture. The truck measures $26$ feet long, $8$ feet high and $7$ feet wide. The volume inside the truck is $1,456~\text{ft}^3$.

  • To calculate the volume, you should recognize that the back of a moving truck is shaped like a rectangular prism.
  • The volume of a rectangular prism is, $V=lwh$.
  • Therefore the volume is $V=26~\text{ft}\cdot 8~\text{ft}\cdot 7~\text{ft}=1,456~\text{ft}^3$.

  2. Isla loves chocolate. She wants to figure out the volume of a chocolate box that resembles a triangular prism. Isla measures the box and finds that the triangular base measures $1.3~\text{in}$ across the bottom and $1.4~\text{in}$ high. The height of the box is $8.3~\text{in}$. Isla calculates the volume to be $7.553~\text{in}^3$.

  • To calculate the volume, you should recognize that a box of Tobblerone chocolate is shaped like a triangular prism.
  • The volume of a triangular prism is, $V=\frac{1}{2}\cdot b\cdot h_{1}\cdot h_{2}$.
  • Therefore the volume is $V=\frac{1}{2}\cdot 1.3~\text{in}\cdot 1.4~\text{in}\cdot 8.3~\text{in}=7.533~\text{in}^3$.

  3. Jordan is sending a birthday present to his Grandmother. He has two boxes to choose from:

  • Box A is $20$ inches long, $8$ inches wide, and $7$ inches high.
  • Box B is $15$ inches long, $12$ inches wide, and $5$ inches high.

  The volume of box A is $1,120~\text{in}^3$ and the volume of box B is $900~\text{in}^3$. Jordan wants the box with the bigger volume so he picks box A.

  • To calculate the volume, you should recognize that a box is shaped like a rectangular prism.
  • The volume of a rectangular prism is, $V=lwh$.
  • The volume for Box A is $V=20~\text{in}\cdot 8~\text{in}\cdot 7~\text{in}=1,120~\text{in}^3$.
  • The volume for Box B is $V=15~\text{in}\cdot 12~\text{in}\cdot 5~\text{in}=900~\text{in}^3$.
  • Since $1,120~\text{in}^3$ is greater than $900~\text{in}^3$, Jordan picks Box A.

• Review the area of shapes and names of 3-D objects.

  Hints

  The name of a 3-D prism generally includes the shape of their base. However, there are a few exceptions.

  To calculate the area of a triangle we need to know the dimensions of its base and height.

  The area of a circle with a radius of $2$ cm is approximately equal to $12.57$ cm$^2$.

  A cone is a solid or hollow object which tapers from a circular base to a point.

  A cylinder is a solid object with:

  • two identical flat ends that are circular
  • and one curved side.

  Solution

  Blue Square*

  • The area of a square with side lengths, $s$, is $A=s^2$.
  • A cube and a square prism are two 3-D shapes with a square base.

  Red Rectangle

  • The area of a rectangle with length $l$ and width $w$ is $A=lw$.
  • A rectangular prism is a 3-D shape with a rectangular base.

  Pink Triangle

  • The area of a triangle with base $b$ and height $h$ is, $A=\frac{1}{2}bh$.
  • A triangular prism is a 3-D shape with a triangular base.

  Yellow Circle

  • The area of a circle with radius $r$ is, $A=\pi r^2$.
  • A cylinder and a cone are two 3-D shapes with a circular base.

• Calculate the volume of the 3-D object.

  Hints

  The volume of a prism is, $V=Bh$, where $B$ represents the area of the base and $h$ represents the height of the prism.

  The base area is represented many different ways, and is dependent on the base shape of the prism.

  Two rectangular prisms that are attached together is called a composite rectangular prism. Multiple volumes of rectangular prisms make up the volume of a composite rectangular prism.

  Solution

  The volume of a prism is $V=Bh$, where $B$ represents the area of the base and $h$ represents the height of the prism.

  1. Composite Rectangular Prism

  The volume of the first rectangular prism with dimensions $7~\text{cm}$ by $2~\text{cm}$ by $5~\text{cm}$ is,

  • $V=[lw]h$
  • $V=7~\text{cm}\cdot 2~\text{cm}\cdot 5~\text{cm}$
  • $V=70~\text{cm}^3$

  The volume of the second rectangular prism with dimensions $4~\text{cm}$ by $4~\text{cm}$ by $2~\text{cm}$ is,

  • $V=[lw]h$
  • $V=4~\text{cm}\cdot 4~\text{cm}\cdot 2~\text{cm}$
  • $V=32~\text{cm}^3$

  The total volume is,

  • $V=70~\text{cm}^3+32~\text{cm}^3=102~\text{cm}^3$

  2. Cylinder

  • $V=[\pi \cdot r^2]\cdot h$
  • $V=3.14 \cdot 3~\text{cm}^2\cdot 9~\text{cm}$
  • $V=254.34~\text{cm}$

  3. Trapezoidal Prism

  • $V=[\frac{1}{2}(b_1+b_2)\cdot h_{1}]\cdot h_{2}$
  • $V=\frac{1}{2}(6~\text{cm}+9~\text{cm})\cdot 5~\text{cm}\cdot 18~\text{cm}$
  • $V=675~\text{cm}^3$

  4. Triangular Prism

  • $V=[\frac{1}{2}\cdot b\cdot h_{1}]\cdot h_{2}$
  • $V=\frac{1}{2}\cdot 20$cm$\cdot 12~\text{cm}\cdot 25~\text{cm}$
  • $V=3,000~\text{cm}^3$