Back To Course

Geometry: High School15 chapters | 160 lessons

Are you a student or a teacher?

Start Your Free Trial To Continue Watching

As a member, you'll also get unlimited access to over 70,000 lessons in math, English, science, history, and more. Plus, get practice tests, quizzes, and personalized coaching to help you succeed.

Free 5-day trial
Your next lesson will play in
10 seconds

Lesson Transcript

Instructor:
*Yuanxin (Amy) Yang Alcocer*

Amy has a master's degree in secondary education and has taught math at a public charter high school.

Watch this video lesson to learn where in the real world you will see cones. Also, you will learn the formulas for finding the surface area and volume of these cones.

When I hear the word 'cones,' I think of ice cream cones, especially the waffle kind. The general shape of these waffle cones for ice cream is that of the cones that we are talking about in this lesson. Mathematically defined, a **cone** is a three-dimensional object with a flat base and one curved side that slants to a tip at the top. Another real-world example of a cone is those orange traffic cones. Can you see how both ice cream cones and the traffic cones have one curved side that slants to a point?

When working with cones, there are three measurements that we need to be concerned about. They are the radius, the height of the cone, and the length of the side from tip to base. We can label these below on a drawing of a cone with *r* for the radius, the distance from the center of the circular base to the edge of the base; *h* for the height of the cone, how tall the cone stands; and *s* for length of the side from tip to base. Our formulas for surface area and volume use these measurements.

Our formula for surface area has two parts, one for the base and one for the curved side. The surface area is the total area of just the surface of our object. The formula for the area of the base is:

*Surface Area Base* = Ï€*r*2

The formula for the curved side is:

*Surface Area Curved Side* = Ï€*rs*

The *s* can be rewritten using the height and radius as:

*s* = âˆš(*r*2 + *h*2)

We need to add both the base and curved side together to get the total surface area, so our complete surface area formula for a cone is:

*Surface Area* = Ï€*r*2 + Ï€*rs*

If our problem gives us the radius and the *s* measurement, the length of the side from tip to base, then we can plug that number directly into the formula for *s*. But if the problem only gives us the height and the radius of the cone, then we would have to rewrite our problem like this:

Surface Area = Ï€*r*2 + Ï€*r* * âˆš(*r*2 + *h*2)

We use this version of the formula if we are only given the height and radius of the cone.

Let's see how this works with a sample problem. We have a cone, and we are told that it has a radius of 3 inches and a height of 10 inches. We see that we are given the *r* and the *h*, but not the *s*. So, this tells me that I need to use the second version of the surface area formula that uses the height instead of the side length. So, I plug in my numbers into this formula; I plug in 3 for *r* and 10 for *h* wherever I see them:

Surface Area = 3.14 * 32 + 3.14 * 3 * âˆš(32 + 102)

Now I go ahead and evaluate to find my answer. I square the 3 and 10 and then I add them together to get 109. I take the square root of this to get 10.44, which I then multiply with the 3 and the 3.14 to get 98.35. I then multiply the other part of 3.14 with the square of 3, which gives me 28.26. I then add the 28.26 and the 98.35 together to get my answer of 126.61 inches squared.

Surface Area = 3.14 * 32 + 3.14 * 3 * âˆš(9 + 100)

Surface Area = 3.14 * 32 + 3.14 * 3 * âˆš(109)

Surface Area = 3.14 * 32 + 3.14 * 3 * 10.44

Surface Area = 3.14 * 32 + 98.35

Surface Area = 3.14 * 9 + 98.35

Surface Area = 28.26 + 98.35

Surface Area = 126.61 inches squared

I remember that area is always squared, so I make sure my answer will always end with my measuring units squared.

Now, if my problem gave me the side length instead, I would go through the same process except I would be using the version of the formula with the *s* instead of the *h*.

For volume, the amount of space inside our object, there is only one version of the formula, which uses the height instead of the side length. The formula for the volume of a cone is:

Volume =

Again, once you have your radius and height, all you need to do is plug these numbers into the formula and evaluate to get your answer.

Let's find the volume of the sample cone that we found the surface area for. Let's see. Our radius for the cone is 3 inches, and our height is 10 inches. We plug these numbers into our formula to get the following.

Volume =

We then evaluate this by squaring the 3 to get 9 and then multiplying that by 3.14 and 10 to get 282.6, which we divide by 3 to get our answer of 94.2 inches cubed.

I recall that volume is always cubed, so I make sure my answer includes my measuring units cubed.

What have we learned? We learned that **cones** are three-dimensional objects with a flat base and one curved side that slants to a tip at the top. In the real world, we see cones such as ice cream waffle cones and traffic cones. The measurements that we are concerned about for cones are the radius, the height, and the length of the side from tip to base.

The formula for finding the surface area of a cone is:

*Surface Area* = Ï€*r*2 + Ï€*rs*

where the *s* can be replaced by: âˆš(*r*2 + *h*2)

if the problem gives you the height instead of the side length.

The formula for volume is:

Volume =

where all you need are the height and radius.

With both of these formulas, all you need is to plug in your appropriate numbers and evaluate to find your answers. Just remember that area is always squared and volume is always cubed.

Finishing this lesson could enable you to:

- Illustrate the shape of a cone and relate it to real-world objects
- Recognize the formulas for surface area and volume of a cone
- Write out the two formulas used to find the surface area

To unlock this lesson you must be a Study.com Member.

Create your account

Are you a student or a teacher?

Already a member? Log In

BackDid you know… We have over 160 college courses that prepare you to earn credit by exam that is accepted by over 1,500 colleges and universities. You can test out of the first two years of college and save thousands off your degree. Anyone can earn credit-by-exam regardless of age or education level.

To learn more, visit our Earning Credit Page

Not sure what college you want to attend yet? Study.com has thousands of articles about every imaginable degree, area of study and career path that can help you find the school that's right for you.

You are viewing lesson
Lesson
6 in chapter 11 of the course:

Back To Course

Geometry: High School15 chapters | 160 lessons

- Planes and the Polyhedron: Definition and Example 3:52
- What Are Platonic Solids? - Definition and Types 4:39
- Prisms: Definition, Area & Volume 6:12
- Pyramids: Definition, Area & Volume 7:43
- What Are Cylinders? - Definition, Area & Volume 5:09
- Cones: Definition, Area & Volume 8:59
- Go to High School Geometry: Geometric Solids

- OAE Prekindergarten Subtests I & II (036/037): Study Guide & Practice
- NES Essential Academic Skills: Study Guide & Practice
- OAE Foundations of Reading (090): Study Guide & Practice
- OSAT School Psychologist Exam (CEOE) (033): Study Guide & Practice
- Praxis Physical Education - Content & Design (5095): Study Guide & Practice
- Exercise & Nutrition
- Early Childhood Development Overview
- Overview of Culture
- Basic Instructional Strategies for Physical Education
- Careers in Healthcare
- AFOQT Cost
- What Does the HESI A2 Nursing Exam Consist of?
- How to Learn Pharmacology for NCLEX
- What Are Considered Higher-Level Questions on the NCLEX?
- How to Study for NCLEx in 2 Weeks
- How Hard Is the ASVAB
- How Long is the HESI A2 Nursing Exam?

- Gender Identity Discrimination in the Workplace: Definition, Laws & Cases
- Teaching Math to Students with Autism
- Aerobic Gram-Positive Bacilli: Characteristics, Types & Examples
- Income Elasticity of Demand in Microeconomics
- What Are the Benefits of Hiring Veterans?
- Practical Application: Creating an Inclusive Workplace Environment for Veterans Infographic
- How to Accommodate Religious Dietary Restrictions for Your Employees
- Veterans in the Workplace: Work Style & Characteristics
- Quiz & Worksheet - Phonics Instruction
- Quiz & Worksheet - Effects of Disabilities on Self & Others
- Quiz & Worksheet - Concept Generalization Teaching Methods
- Quiz & Worksheet - Parental Involvement in IEPs
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- College Macroeconomics: Tutoring Solution
- AEPA Middle Grades General Science: Practice & Study Guide
- NES Elementary Education Subtest 2 (103): Practice & Study Guide
- GACE Physics (530): Practice & Study Guide
- CLEP American Literature: Study Guide & Test Prep
- AFOQT: Instrument Knowledge
- Foreign Exchange Market & Purchasing Power
- Quiz & Worksheet - Artists in Renaissance Society
- Quiz & Worksheet - Themes in Renaissance Art
- Quiz & Worksheet - Impact of a Poem or Drama's Structure on Its Meaning
- Quiz & Worksheet - Impact of Politics on Greek Art & Culture
- Quiz & Worksheet - The U.S.'s Religious Makeup

- Food Allergies & Intolerances: Differences & Treatments
- Echo Reading Activities
- What is the AFQT Percentile Score?
- What To Do If Your School Doesn't Accept Study.com Credit
- High School Summer Reading List
- Expelled from School: Next Steps
- Globalization & International Management: Assignment 1
- What is a Lexile Score?
- Persuasive Writing Prompts: 3rd Grade
- What is the International Baccalaureate Primary Years Program?
- Volleyball Lesson Plan
- Money Management Lesson Plan

- Tech and Engineering - Videos
- Tech and Engineering - Quizzes
- Tech and Engineering - Questions & Answers

Browse by subject