{"id":2826,"date":"2023-05-15T18:55:47","date_gmt":"2023-05-15T18:55:47","guid":{"rendered":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/?post_type=chapter&p=2826"},"modified":"2025-08-26T03:57:18","modified_gmt":"2025-08-26T03:57:18","slug":"volume-and-surface-area-learn-it-4","status":"web-only","type":"chapter","link":"https:\/\/content.one.lumenlearning.com\/quantitativereasoning\/chapter\/volume-and-surface-area-learn-it-4\/","title":{"raw":"Volume and Surface Area: Learn It 4","rendered":"Volume and Surface Area: Learn It 4"},"content":{"raw":"

Finding the Volume of a Cone<\/h2>\r\n

The first image that many of us have when we hear the word \u2018cone\u2019 is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In geometry, a 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.<\/p>\r\n

\r\n[caption id=\"attachment_9099\" align=\"aligncenter\" width=\"300\"]\"A Figure 1. The cone's height, h, and radius, r, is labeled[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

Earlier in this section, we saw that the volume of a cylinder is [latex]V=\\pi{r}^{2}h[\/latex]. We can think of a cone as part of a cylinder. The image 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>\r\n

\r\n[caption id=\"attachment_9100\" align=\"aligncenter\" width=\"225\"]\"A Figure 2. Think of the cone as part of a cylinder[\/caption]\r\n<\/center>\r\n

 <\/p>\r\n

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>\r\n

[latex]V = \\frac{1}{3}\\textcolor{orange}{B}h[\/latex]<\/center>\r\n

 <\/p>\r\n

Since the base of a cone is a circle, we can substitute the formula of area of a circle, [latex]\\pi{r}^{2}[\/latex], for [latex]B[\/latex] to get the formula for volume of a cone.<\/p>\r\n

[latex]V = \\frac{1}{3}\\textcolor{orange}{\\pi r^2}h[\/latex]<\/center>\r\n

 <\/p>\r\n

In this book, we will only find the volume of a cone, and not its surface area.<\/p>\r\n

\r\n
\r\n

volume of a cone<\/h3>\r\n\r\nFor a cone with radius [latex]r[\/latex] and height [latex]h[\/latex].
\"A<\/center>\r\n

 <\/p>\r\n<\/div>\r\n<\/section>\r\n

Find the volume of a cone with height [latex]6[\/latex] inches and radius of its base [latex]2[\/latex] inches.[reveal-answer q=\"57883\"]Show Solution[\/reveal-answer] [hidden-answer a=\"57883\"]\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n\r\n
Step 1. Read<\/strong> the problem. Draw the figure and label it with the given information.<\/td>\r\n\"A<\/td>\r\n<\/tr>\r\n
Step 2. Identify<\/strong> what you are looking for.<\/td>\r\nthe volume of the cone<\/td>\r\n<\/tr>\r\n
Step 3. Name.<\/strong> Choose a variable to represent it.<\/td>\r\nlet [latex]V[\/latex] = volume<\/td>\r\n<\/tr>\r\n
Step 4. Translate.<\/strong> Write the appropriate formula. Substitute. (Use [latex]3.14[\/latex] for [latex]\\pi [\/latex] )<\/td>\r\n[latex]V=\\Large\\frac{1}{3}\\normalsize\\pi {r}^{2}h[\/latex] [latex]V\\approx \\Large\\frac{1}{3}\\normalsize 3.14{\\left(2\\right)}^{2}\\left(6\\right)[\/latex]<\/td>\r\n<\/tr>\r\n
Step 5. Solve.<\/strong><\/td>\r\n[latex]V\\approx 25.12[\/latex]<\/td>\r\n<\/tr>\r\n
Step 6. Check:<\/strong> We leave it to you to check your calculations.<\/td>\r\n <\/td>\r\n<\/tr>\r\n
Step 7. Answer<\/strong> the question.<\/td>\r\nThe volume is approximately [latex]25.12[\/latex] cubic inches.<\/td>\r\n<\/tr>\r\n<\/tbody>\r\n<\/table>\r\n\r\n[\/hidden-answer]<\/section>\r\n
[ohm2_question hide_question_numbers=1]6987[\/ohm2_question]<\/section>","rendered":"

Finding the Volume of a Cone<\/h2>\n

The first image that many of us have when we hear the word \u2018cone\u2019 is an ice cream cone. There are many other applications of cones (but most are not as tasty as ice cream cones). In geometry, a 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.<\/p>\n

\n
\"A
Figure 1. The cone’s height, h, and radius, r, is labeled<\/figcaption><\/figure>\n<\/div>\n

 <\/p>\n

Earlier in this section, we saw that the volume of a cylinder is [latex]V=\\pi{r}^{2}h[\/latex]. We can think of a cone as part of a cylinder. The image 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>\n

\n
\"A
Figure 2. Think of the cone as part of a cylinder<\/figcaption><\/figure>\n<\/div>\n

 <\/p>\n

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>\n

[latex]V = \\frac{1}{3}\\textcolor{orange}{B}h[\/latex]<\/div>\n

 <\/p>\n

Since the base of a cone is a circle, we can substitute the formula of area of a circle, [latex]\\pi{r}^{2}[\/latex], for [latex]B[\/latex] to get the formula for volume of a cone.<\/p>\n

[latex]V = \\frac{1}{3}\\textcolor{orange}{\\pi r^2}h[\/latex]<\/div>\n

 <\/p>\n

In this book, we will only find the volume of a cone, and not its surface area.<\/p>\n

\n
\n

volume of a cone<\/h3>\n

For a cone with radius [latex]r[\/latex] and height [latex]h[\/latex].<\/p>\n

\"A<\/div>\n

 <\/p>\n<\/div>\n<\/section>\n

Find the volume of a cone with height [latex]6[\/latex] inches and radius of its base [latex]2[\/latex] inches.<\/p>\n