Back To Course

Math 104: Calculus14 chapters | 115 lessons | 11 flashcard sets

Watch short & fun videos
**
Start Your Free Trial Today
**

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:
*Heather Higinbotham*

Squares pegs = square holes. Triangular pegs = triangular holes. But where does a sphere go? In this lesson, review volumes of common shapes while contrasting a sphere and a cylinder - after all, they both go into the circular hole... right?

Let's take a few minutes to review the **volumes of shapes**. These get quite important in calculus problems because we often use calculus to do things like optimize areas, optimize volumes - you know, how much shampoo should you put in a shampoo bottle to make the most money? - that kind of thing. I also like to call this the square peg in square hole problem. And this is because of how I classify shapes.

Let's start with **2D shapes**. So, now instead of looking at volume, let's look at the **area**. So, if you have a square with a side length *s*, then the area of the square is *s* ^2. The **perimeter**, that is the length that borders the shape, is 4 * *s*. Now, a square is just a fancy type of rectangle. A rectangle has some height, *h*, and some width, *w*, so the area of a rectangle is *wh* - the width times the height. You can find the length of this line that surrounds the rectangle, or the perimeter, as 2 * *w* + 2 * *h* because we're adding up *w* + *h* + *w* + *h* when we go around the rectangle.

Now, the second of our square peg in square hole area is the triangle. So, let's take a look at two different types of triangles. We've got this guy here, and we've got this guy here. Now both of these have a base that's the width of the bottom part of the triangle, and it's always going to be along some side. We're going to call that *b*. Now the triangles also have some height. It's really important that the height be measured as being perpendicular to the base. So, for some triangles, such as this one here, we need to measure the height actually outside of the base because we need this height to be straight up from the base, to be **perpendicular** from the base. In these cases, the area is just 1/2 *bh*. Now, I know this is all a review, so I won't explain why.

Now, the last of my three types of shapes in my square peg, square hole is the circle. We usually classify a circle by some **radius**, *r*, measured from the center of the circle straight out to the edge of the circle. So, this radius is the same no matter where along the circle you measure it. The area of a circle like this is *pi r* ^2.

Okay. Square peg - square hole. Rectangular peg - rectangular hole. Triangular peg - triangular hole. Circular peg - circular hole. Well, that's fine but these are all 3D shapes. So this brings us to my two ways to classify 3D shapes.

The first, shapes that are the same from the top to the bottom. So, here we go with the pegs. We've got a can. So, a can will fit in this circular hole and it will fit the same, no matter if it is the beginning of the can or the end. There's no difference. It's just a cylinder. A cube is another example. So, here the bottom of the cube - this cross-section here - looks just the same as the top. Prism - same thing, except now your cross-section looks like a triangle.

The second classification for shapes, are shapes that aren't the same from the top to the bottom. So, these are things like a square pyramid, where we've got a square on the bottom and a point at the top, or a regular pyramid with a triangle on the bottom and a point on the top. If I put the square pyramid into my hole, it will be a square at the bottom, but at the top it will just be a little point going through that hole. Same thing with the triangular pyramid. A sphere is another example. If you slice through a sphere at the top, you're going get a little, teeny, tiny circle. At the very top you're going to get no circle whatsoever. It's just going to be a point. In the middle, the circle's going to be big. So imagine you're taking an onion, and if you cut straight down, perpendicular to the cutting board, you're going to end up with a lot of differently sized circles.

So, what does this mean in terms of finding the volume? Well, for anything that has the same shape on the top and the bottom and all the way through, like the can, the cube and the prism, the volume is just the height times the area of the base. So, here we've got our can. We've got height. The base area is just *pi r* ^2. So, there's my base. So, the volume is *h*(*pi r* ^2). Say now I've got this box, this trunk of sorts, with a base that is a square. So, now my base area is *s* ^2 because that's the area of a square, and my volume is going to be the base area, *s* ^2 times *h*. So my volume is *h*(*s* ^2). So, you can do this for anything where the cross-section is the same from the top to the bottom. Say your base area is an E. It looks like a big E. Same thing. Base area times the height will give you the volume.

So, what about things that change? Let's look at some of those. For example, a sphere: the volume of a sphere is 4/3 *pi r* ^3. A hemisphere is just half a sphere, and so the volume of a hemisphere is 1/2 (4/3 *pi r* ^3), which is also 2/3 *pi r* ^3.

Pyramids and cones are treated very similarly to one another. So, each one has a height, and the volume is the h/3 (area of the base). You can remember this for both of those. So, the area of the base of a cone, obviously, is the area of this circle, and for a pyramid it's going to be, in this case, a triangle, sometimes a square, but that's going to be the area of the base.

So, let's recap. When you're trying to find the volume of something just remember if it's a square peg in a square hole. If the base cross-section is the same as the cross-section throughout the entire volume then your volume is the base area, or one of those cross-sections, times the height. If it's not the same, then you need to remember one of the other rules. For example, for cones, your volume is h/3 (base area). For circles, just remember the equation for the volume of a sphere: 4/3 *pi r* ^3.

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

Create
your account

Already a member? Log In

BackDid you know… We have over 95 college courses that prepare you to earn credit by exam that is accepted by over 2,000 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
2 in chapter 3 of the course:

Back To Course

Math 104: Calculus14 chapters | 115 lessons | 11 flashcard sets

- Reducing Stress for Supervisors
- Team Building Skills for Supervisors
- Stakeholder Agility for Leaders
- Developing Adaptable Teams & Employees
- Effective Delegation Skills for Supervisors
- Managing Stress With a Supportive Work Environment
- Managing Employee Stress in the Workplace
- Currency Changes in International Finance
- The Balance of Payments in International Finance
- Defining, Understanding & Managing Stress
- How to Request a CLEP Transcript
- CLEP Exam Dates & Testing Center Locations
- CLEP Scoring System: Passing Scores & Raw vs. Scaled Score
- Continuing Education Opportunities for Molecular Biology Technologists
- WV College & Career Readiness Standards for Social Studies
- Common Core State Standards in Ohio
- Resources for Assessing Export Risks

- The Role of the Leader During Organizational Change
- Amino Acid Side Chains: Function & Examples
- Employment Issues: Law, Selection & Performance Appraisal
- International Tourism & Travel Organizations
- Relational Expressions in C Programming: Types & Examples
- Examples of Stressors in the Workplace
- Difference Between Chiral & Enantiomers
- Options & Capital Budgeting: Timing, Strategy & Planning
- Quiz & Worksheet - Sweatshops and Child Labor
- Quiz & Worksheet - Edward Chace Tolman & Purposive Behaviorism
- Quiz & Worksheet - Gender Discrimination Laws & Workplaces
- Quiz & Worksheet - Electric Fields Calculations
- Quiz & Worksheet - What is Thermal Conductivity?
- Tourism Marketing Flashcards
- Tourism Economics Flashcards

- MTTC Economics: Practice & Study Guide
- NMTA Essential Academic Skills Subtest Math: Practice & Study Guide
- NYSTCE Chemistry: Practice and Study Guide
- Emotional Intelligence: Help & Review
- MTTC Mathematics (Secondary): Practice & Study Guide
- MTEL Middle School Math/Science: Thermodynamics in Chemistry
- TACHS: Factoring
- Quiz & Worksheet - Importance of HR Development
- Quiz & Worksheet - Uncertainty Reduction
- Quiz & Worksheet - Star-Crossed Lovers in Shakespeare
- Quiz & Worksheet - Types of Employee Incentives
- Quiz & Worksheet - Incapacitation in Criminal Justice

- What Is Collaboration in the Workplace? - Definition, Benefits & Examples
- Daniel Inouye: Biography & Medal of Honor
- Harlem Renaissance Lesson Plan
- Praxis Tests in Hawaii
- New Deal Lesson Plan
- Oregon Trail Lesson Plan
- Books for Teaching All About Me
- What is a Distance Learning Course?
- Behavior Rubric Examples
- Louisiana Purchase Lesson Plan
- How to Ace a Phone Interview
- How to Write a Good College Essay

Browse by subject