<h2>Finding the Volume of Cones</h2>
<p>The first image that many of us have when we hear the word ‘cone’ is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In this section, we will see how to find the volume of a cone.</p>
<p>In geometry, a <strong>cone</strong> is a solid figure with one circular base and a vertex. The height of a cone is the distance between its base and the vertex.The cones that we will look at in this section will always have the height perpendicular to the base. See the figure below.</p>
<div class="wp-caption alignnone" style="width: 136px"><img alt="An image of a cone is shown. The top is labeled vertex. The height is labeled h. The radius of the base is labeled r." height="132" src="https://data.ulearngo.com/assets/4596b576-0916-47ec-9573-b9dd226dfe3f" width="136"/><p class="wp-caption-text">The height of a cone is the distance between its base and the vertex.</p></div>
<p>Earlier in this section, we saw that the volume of a cylinder is \(V=\text{π}{r}^{2}h.\) We can think of a cone as part of a cylinder. The figure below shows a cone placed inside a cylinder with the same height and same base. If we compare the volume of the cone and the cylinder, we can see that the volume of the cone is less than that of the cylinder.</p>
<div class="wp-caption alignnone" style="width: 104px"><img alt="An image of a cone is shown. There is a cylinder drawn around it." height="138" src="https://data.ulearngo.com/assets/1ee166de-3b32-42b6-b763-fdc468bad208" width="104"/><p class="wp-caption-text">The volume of a cone is less than the volume of a cylinder with the same base and height.</p></div>
<p>In fact, the volume of a cone is exactly one-third of the volume of a cylinder with the same base and height. The volume of a cone is</p>
<p><img alt="The formula V equals one-third times capital B times h is shown." src="https://data.ulearngo.com/assets/08f4ba58-2e77-472e-a19e-a458e3729149"/></p>
<p>Since the base of a cone is a circle, we can substitute the formula of area of a circle, \(\text{π}{r}^{2}\) , for \(B\) to get the formula for volume of a cone.</p>
<p><img alt="The formula V equals one-third times pi times r squared times h is shown." src="https://data.ulearngo.com/assets/24bd318d-91d2-4866-8f62-e7d7ac675b6c"/></p>
<p>In this book, we will only find the volume of a cone, and not its surface area.</p>
<div class="ns-tutorial">
<div class="ns-school-defn2">
<h3>Definition: Volume of a Cone</h3>
<p>For a cone with radius \(r\) and height \(h\).</p>
<img alt="An image of a cone is shown. The height is labeled h, the radius of the base is labeled r. Beside this is Volume: V equals one-third times pi times r squared times h." src="https://data.ulearngo.com/assets/89565631-6252-477d-99a6-4208184fd21e"/></div>
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<div class="ns-school-emp">
<h2>Example</h2>
<div class="ns-school-defn">
<div data-type="example">
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<div>
<p>Find the volume of a cone with height \(6\) inches and radius of its base \(2\) inches.</p>
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<h3>Solution</h3>
<table data-label="" summary="The text reads, “Step 1. Read the problem.” Beside this is an image of a cone. The radius is labeled 2, the height is labeled 6.">
<tbody>
<tr>
<td>Step 1. <strong>Read</strong> the problem. Draw the figure and label it

<div> </div>

with the given information.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/e96baefb-7e97-4b2c-bdc3-eb8691490da3"/></td>
</tr>
<tr>
<td>Step 2. <strong>Identify</strong> what you are looking for.</td>
<td>the volume of the cone</td>
</tr>
<tr>
<td>Step 3. <strong>Name.</strong> Choose a variable to represent it.</td>
<td>let <em>V</em> = volume</td>
</tr>
<tr>
<td>Step 4. <strong>Translate.</strong>
<div> </div>

Write the appropriate formula.

<div> </div>

Substitute. (Use 3.14 for \(\pi \))</td>
<td>
<div> </div>

\(V=\frac{1}{3}\phantom{\rule{1em}{0ex}}\pi \phantom{\rule{1.9em}{0ex}}{r}^{2}\phantom{\rule{1.7em}{0ex}}h\)

<div> </div>

\(V\approx \frac{1}{3}\phantom{\rule{0.7em}{0ex}}3.14\phantom{\rule{1em}{0ex}}{\left(2\right)}^{2}\phantom{\rule{1em}{0ex}}\left(6\right)\)</td>
</tr>
<tr>
<td>Step 5. <strong>Solve.</strong></td>
<td>\(V\approx 25.12\)</td>
</tr>
<tr>
<td>Step 6. <strong>Check:</strong> We leave it to you to check your

<div> </div>

calculations.</td>
<td> </td>
</tr>
<tr>
<td>Step 7. <strong>Answer</strong> the question.</td>
<td>The volume is approximately 25.12 cubic inches.</td>
</tr>
</tbody>
</table>
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<div class="ns-school-emp">
<h2>Example</h2>
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<div data-type="example">
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<p>Marty’s favorite gastro pub serves french fries in a paper wrap shaped like a cone. What is the volume of a conic wrap that is \(8\) inches tall and \(5\) inches in diameter? Round the answer to the nearest hundredth.</p>
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<div>
<h3>Solution</h3>
<table data-label="" summary="Step 1 says, “Read the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.” The word “read” is in bold. Beside this is an image of a cone. The diameter of the circle at the top of the cone is labeled 5. The height of the cone is labeled 8. Step 2 says, “Identify what you are looking for.” The word “identify” is in bold. Beside this it says, “the volume of the cone.” Step 3 says, “Name. Choose a variable to represent it.” The word “name” is in bold. Beside this it says, “let V equal volume.” Step 4 says, “Translate. Write the appropriate formula. Substitute. (Use 3.14 for pi, and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.)” The word “translate” is in bold. Beside this is V equals one-third times pi times r squared times h. Below that is V is approximately one-third times 3.14 times 2.5 in. squared times 8 in. Step 5 says, “Solve the equation.” The word “solve” is in bold. Beside this is V is approximately 52.33. Step 6 says, “Check,” in bold. Then “We leave it to you to check your arithmetic.” Step 7 says, “Answer the question.” The word “answer” is in bold. The answer is given as “The volume of the wrap is approximately 52.33 cubic inches.”">
<tbody>
<tr>
<td>Step 1. <strong>Read</strong> the problem. Draw the figure and label it with the given information. Notice here that the base is the circle at the top of the cone.</td>
<td><img alt="." src="https://data.ulearngo.com/assets/2e964a1b-ef26-42af-9d9f-491f940d139b"/></td>
</tr>
<tr>
<td>Step 2. <strong>Identify</strong> what you are looking for.</td>
<td>the volume of the cone</td>
</tr>
<tr>
<td>Step 3. <strong>Name.</strong> Choose a variable to represent it.</td>
<td>let <em>V</em> = volume</td>
</tr>
<tr>
<td>Step 4. <strong>Translate.</strong> Write the appropriate formula. Substitute. (Use 3.14 for \(\pi \), and notice that we were given the distance across the circle, which is its diameter. The radius is 2.5 inches.)</td>
<td>
<div> </div>

\(V=\frac{1}{3}\phantom{\rule{1em}{0ex}}\pi \phantom{\rule{2.3em}{0ex}}{r}^{2}\phantom{\rule{2em}{0ex}}h\)

<div> </div>

\(V\approx \frac{1}{3}\phantom{\rule{0.7em}{0ex}}3.14\phantom{\rule{1em}{0ex}}{\left(2.5\right)}^{2}\phantom{\rule{1em}{0ex}}\left(8\right)\)</td>
</tr>
<tr>
<td>Step 5. <strong>Solve.</strong></td>
<td>\(V\approx 52.33\)</td>
</tr>
<tr>
<td>Step 6. <strong>Check:</strong> We leave it to you to check your calculations.</td>
<td> </td>
</tr>
<tr>
<td>Step 7. <strong>Answer</strong> the question.</td>
<td>The volume of the wrap is approximately 52.33 cubic inches.</td>
</tr>
</tbody>
</table>
</div>
</div>
</div>
</div>
</div>
<h3>Optional Video: Volume of a Cone</h3>
<p><iframe allowfullscreen="allowfullscreen" frameborder="0" height="386" src="https://www.youtube.com/embed/7Y0ZMnCcVGs?rel=0&amp;modestbranding=1&amp;cc_lang_pref=en&amp;cc_load_policy=1" width="686"><span class="mce_SELRES_start" data-mce-type="bookmark" style="display: inline-block; width: 0px; overflow: hidden; line-height: 0;">﻿</span></iframe></p>

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Finding the Volume of Cones

Mathematics studies measurement, relationships, and properties of quantities and sets, using numbers and symbols. Arithmetic, algebra, geometry, and calculus are popular topics in mathematics.

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Discover problem-solving strategies for word problems, solving number and coin word problems, understanding properties of angles and triangles, and finding volume and surface area of shapes in Math Models and Geometry I tutorial.


Finding the Volume of Cones

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