Volume of a Cone

Volume of a Cone exercise

• Understand what the volume of a cone measures.

  Hints

  A cone is a three-dimensional shape, such as this party hat! Some other common cones are:

  • ice cream cones
  • traffic cones
  • funnels

  Like any three-dimensional shape, the volume measures the interior 3D space.

  Solution

  The volume of a cone measures the amount of space inside the cone.

• Using a formula to find the volume of a cone.

  Hints

  The radius is the distance from the center of a circle to the outside.

  The height of a cone measures the distance from the center of the circle to the tip of the cone. This line of measurement must be perpendicular to the base.

  Substituting a value for a variable means replacing the variable with a specific number in an equation or expression, which allows you to solve or simplify it.

  For example, if we know that $x=4$, and we need to evaluate $x - 6$, we would then substitute to write $\bf{4} - 6$.

  Solution

  The radius of the cone is $\bf{2\:cm}$.

  The height of the cone is $\bf{4\:cm}$.

  The formula used to find the volume of the cone is $\bf{V=\frac{1}{3} \pi r^2h}$.

  Once the values for the radius and height are identified, substitute them for the appropriate variables, and write the equation: $\bf{V=\frac{1}{3} \pi (2^2)(4)}$.

• Volume of a Cone.

  Hints

  To use a formula, substitute in the values for the corresponding variable.

  For example, the cone seen here has the following measurements:

  $r=3\:cm$

  $h=2\:cm$

  The formula to find the volume is: $V=\frac{1}{3}\pi\: r^2h$

  When the values are substituted, the equation is: $V=\frac{1}{3}\pi \:(3^2)2$.

  Solution

  Substitute the values of the radius and height into the formula, $V=\frac{1}{3}\pi \:r^2h$, for $r$ and $h$

• Find the volume of a cone.

  Hints

  The formula for volume of a cone is $V=\frac{1}{3}\pi r^2h$.

  In terms of $\pi$ means to not calculate $\pi$ and leave it as the symbol in the answer instead.

  To calculate the volume of this cone (in terms of $\pi$), first substitute in the known values.

  $V=\frac{1}{3} \pi \:r^2h$

  $V=\frac{1}{3} \pi \:(6^2)(15)$

  Then calculate the product of all the numbers except pi.

  $\frac{1}{3}(6^2)(15)=180$

  Lastly, the pi symbol can be written next to the product, to show the exact value for the volume.

  $V=180\pi\:m^3$

  Solution

  $r=2$

  $h=6$

  $V=\frac{1}{3}\pi r^2 h$

  $V=\frac{1}{3} \pi (2^2) (6)$

  $\frac{1}{3}(2^2)(6)=8$

  $V=8\pi\:in^3$

• Identify the variables in the formula for the volume of a cone.

  Hints

  Remember, this problem is about the volume of a cone. Which measurements are important to finding this value?

  Look at the cone. What measurements do the $h$ and $r$ represent?

  Solution

  The variables and symbols in the Volume of a Cone formula are:

  V = volume

  r = radius

  h = height

  $\pi$ = pi

• Calculate the volume of a cone using the formula.

  Hints

  Formula for the volume of a cone: $V=\frac{1}{3}\pi r ^2 h$

  If you do not have a calculator handy, you can use the approximate value of pi, $3.14...$.

  Remember, the $r$ is the radius of the cone, and the $h$ is the height of the cone.

  Make sure to take your time and show your work to find the volume.

  • Identify the radius and height.
  • Substitute these values in the formula $V=\frac{1}{3}\pi r^2 h$.
  • Calculate the volume and round to the nearest tenth.

  Solution

  The volume for each cone:

  V $\approx$ $804.2\:cm^3$

  V $\approx$ $212.1\:cm^3$

  V $\approx$ $83.8\:cm^3$

  Steps to solve the first one:

  $V=\frac{1}{3}\pi r ^2 h$

  $r=8$

  $h=12$

  $V=\frac{1}{3}\pi (8^2) (12)$

  $V=256 \pi$

  $V \approx 804.2\:cm^3$