Back To Course

Supplemental Math: Study Aid1 chapters | 19 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 75,000 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:
*Sarah Spitzig*

Sarah has taught secondary math and English in three states, and is currently living and working in Ontario, Canada. She has recently earned a Master's degree.

In this lesson, you use general and specific formulas to learn how to find the surface area of three-dimensional shapes, such as cubes, prisms, spheres, cones and cylinders.

A **three-dimensional shape** is a solid shape that has height and depth. For example, a sphere and a cube are three-dimensional, but a circle and a square are not.

A **prism** is a three-dimensional shape that has non-curved sides. A cube is a prism, but a sphere is not. A prism has a pair of congruent sides, called bases, like the cube, triangular prism and the rectangular prism. Don't confuse a prism with a pyramid, which only has one base.

Notice that the prisms each have a pair of congruent bases.

The **surface area** of a three-dimensional shape is the sum of all of the surface areas of each of the sides. I like to think of the shape as a birthday present and the surface area as the wrapping paper. If we carefully took the wrapping paper off of the present and added up each side, the total would be the surface area of the shape.

When we are finding the surface area of a 3-D shape, think of it as unfolding the shape, or flattening it out, and then finding the area of each side. When we add all of these areas up, we have the surface area. There are two types of 3-D shapes we will need to find the surface area of - prisms and non-prisms.

When we are looking for the surface area of a prism, we add all of the areas together to find the total. Another way to find the area of a prism is to find the perimeter of the base and multiply by the height.

*SA (of a prism) = (perimeter of the base) * h + (area of the bases)*

In order to find the area of a 3-D shape, we must know how to find the area of the basic shapes that make up the sides of the 3-D shape. Here is a list of basic shape formulas to help with finding the surface area of the 3-D shapes:

When we are finding the surface area of a prism, we need to find the area of each side, which is one of the basic shapes listed above, and then add all of the areas together to find the total.

There are specific formulas for 3-D shapes that are not prisms. When we are looking for the surface area of a non-prism, like a sphere, cylinder, pyramid and other non-prisms, they each have their own formula, as shown in this table:

First, we need to identify the shape so we know which formula to use. This is a triangular prism since there is a pair of congruent bases. Next, we need to find the area of each side. Since this is a triangular prism, it is made up of two triangles (as the bases) and three rectangles (as the sides).

First, look at the triangular bases. To find the area of a triangle, use:

*A = ( b * h) / 2*

*A = (3 * 4) / 2 = 6*

Now look at the rectangular sides. Since the area of a rectangle is *A* = *l* * *w*, find each side.

*A = 8 * 5 = 40*

*A = 8 * 4 = 32*

*A = 3 * 8 = 24*

Our last step is to add them all up. *A = 6 + 6 + 40 + 32 + 24 = 108 square units*

Another way to find the surface area is to use the formula:

*SA = (perimeter of the base) * h + (area of the bases)*

Since this is a prism, this formula will work.

Since the base is the triangle, we find the perimeter, or distance around it, by adding the sides together.

*P = 5 + 4 + 3 = 12*

Now multiply 12 by the height, which is 96.

Now find the area of the triangular bases. To find the area of a triangle, use:

*A= (b * h) / 2*

*A = (3 * 4) / 2 = 6*

Finally, add them together to find the surface area:

*SA = 96 + 6 + 6 = 108 square units*

First, identify the shape so we know which formula to use. Since this a cylinder, which is a special shape with its own formula, we should first write it out.

*SA = 2(pi)r2 + 2(pi)rh*. Let pi be the number 3.14.

Next, plug in the numbers for pi = 3.14, r = 3 and h = 6.

*SA = (2 x 3.14 x 32) + (2 x 3.14 x 3 x 6)*

Then add them together to get the surface area of the cylinder.

*56.52 + 113.04 = 169.56 square units*

When finding the surface area of a three-dimensional shape, you first identify the shape in order to decide which formula to use. If it is a prism, you have two choices: find the area of each side using the area formulas for each specific shape that makes up each side for a prism and sum the sides to get the total surface area, or use the formula *SA = (perimeter of the base)*h + (area of the bases)*. If it is a not a prism, use one of the special formulas for a non-prism.

Once you are done with this lesson you should be able to:

- Recall the surface area formulas for non-prisms and prisms
- Calculate the surface area of a 3-D shape

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
4 in chapter 1 of the course:

Back To Course

Supplemental Math: Study Aid1 chapters | 19 lessons

- Less Than Symbol in Math: Problems & Applications 4:10
- What are 2D Shapes? - Definition & Examples 4:35
- Trapezoid: Definition, Properties & Formulas 3:58
- What is Surface Area? - Definition & Formulas 5:56
- Universal Set in Math: Definition, Example & Symbol 6:03
- Complement of a Set in Math: Definition & Examples 5:59
- Zero Exponent: Rule, Definition & Examples 4:32
- Quotient Of Powers: Property & Examples 4:58
- What is Simplest Form? - Definition & How to Write Fractions in Simplest Form 5:49
- What is Slope? - Definition & Formulas 7:10
- Skewed Distribution: Examples & Definition 5:09
- Change Of Base Formula: Logarithms & Proof 4:54
- Transformations in Math: Definition & Graph 6:27
- What is Translation in Math? - Definition, Examples, & Terms 4:23
- Fixed Interval: Examples & Definition 4:00
- Scatterplot and Correlation: Definition, Example & Analysis 7:48
- Dilation in Math: Definition & Meaning 5:30
- Simplifying Fractions: Examples & Explanation 4:44
- Go to Overview of Math Concepts

- GRE Information Guide
- Computer Science 310: Current Trends in Computer Science & IT
- Earth Science 105: Introduction to Oceanography
- Computer Science 331: Cybersecurity Risk Analysis Management
- Computer Science 336: Network Forensics
- Natural Resources & Environmental Impact
- World Literature: Drama Since the 20th Century
- Visual Art Since the 18th Century
- World Literature: Drama Through the 19th Century
- Defamation, Libel & Slander
- Study.com GRE Scholarship: Application Form & Information
- PHR Prep Product Comparison
- ILTS Prep Product Comparison
- CTEL Prep Product Comparison
- TASC Prep Product Comparison
- FSA Prep Product Comparison
- SHSAT Prep Product Comparison

- Promoting Parent Involvement in Student Literacy
- Special Populations in Education: Definition & Examples
- Strategies for Teaching Children in Foster Care
- Theodor Schwann: Discovery, Cell Theory & Contributions
- Cash Management Strategies: Spending & Saving
- Cloud Content Delivery Network Services: Features & Providers
- A. A. Milne Lesson Plan
- Practical Application: Using Vivid Language in Public Speaking
- Quiz & Worksheet - Assessments that Measure Accuracy, Rate & Prosody
- Quiz & Worksheet - Measurable Learning Objectives
- Quiz & Worksheet - Java Max & Min Value in Java
- Quiz & Worksheet - Gathering Data from Numerous Assessments
- Quiz & Worksheet - Using Q&A for Reading Comprehension
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- GACE Middle Grades Reading (012): Practice & Study Guide
- Business Ethics: Skills Development & Training
- 9th Grade English: Homeschool Curriculum
- Introduction to Financial Accounting: Certificate Program
- Gerontology for Teachers: Professional Development
- Mathematical Reasoning & Problem-Solving: Help and Review
- Business Law Terms
- Quiz & Worksheet - What is a URL?
- Quiz & Worksheet - Mutations - DNA Errors
- Quiz & Worksheet - Impact of Catastrophic Events on Life on Earth
- Quiz & Worksheet - Encouraging Physical Activity Among Students & the Community
- Quiz & Worksheet - Practice Writing Triple Digit Numbers in Binary

- What is a Math Concept?
- Track & Field Events: Types & Training
- Volleyball Lesson Plan
- STAR Reading Test Scores
- Computer Projects for Kids
- NBCRNA Recertification Requirements
- Do Homeschoolers Have to take Standardized Tests?
- How to Prep for the NYS English Regents Exam
- 8th Grade Math Games
- Common Core State Standards in Idaho
- Free English Language Courses
- Integrated Physics & Chemistry (IPC) Curriculum Overview

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

Browse by subject