Back To Course

High School Geometry: Homework Help Resource13 chapters | 142 lessons

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 55,000 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:
*Jennifer Beddoe*

A pyramid is a 3-dimensional geometric shape formed by connecting all the corners of a polygon to a central apex. This lesson will define what a pyramid is and discuss the different types of pyramids and formulas surrounding them.

A **pyramid** is a 3-dimensional shape whose base is a polygon. Each corner of a polygon is attached to a singular apex, which gives the pyramid its distinctive shape. Each base edge and the apex form a triangle.

There are many types of pyramids. Most often, they are named after the type of base they have. Let's look at some common types of pyramids below.

**Triangular pyramid** has a triangle as its base:

**Square pyramid** has a square as its base:

**Pentagonal pyramid** has a pentagon as its base:

This list could go on and on (hexagonal pyramid, heptagonal pyramid, etc). There are also a few special names for pyramids that you should know.

**Right pyramid** - the apex of the pyramid is directly above the center of its base:

**Oblique pyramid** - the apex of the pyramid is not directly above the center of its base:

**Regular pyramid** - the base of this pyramid is a regular polygon:

**Irregular pyramid** - this type of pyramid has an irregular polygon as its base:

There are formulas that can be used to find both the surface area and the volume of a pyramid. The **surface area** of a pyramid is the total area of all the surfaces that the pyramid has. To that end, the formula for finding the surface area when all of the side faces are the same is:

SA = (base area) + (1/2) * (perimeter) * (slant height)

The **base area** is the area of the base and can be determined based on what figure the base is. For example, if the base of the pyramid is a square, the formula for finding the area is:

A = *s*^2

The **perimeter** is the distance around the base of the pyramid. The **slant height** is the diagonal height from the center of one of the base edges to the apex.

If the pyramid has side faces that differ from each other (like in the case of an irregular pyramid), then the surface area equation is:

SA = (base height) + (lateral area)

In this case, you must take each side of the pyramid separately (including the base), find the areas, and then just add them together.

The volume of a pyramid can be found using this formula:

V = (1/3) * (base area) * (height)

The base area is, again, just the area of the base of the pyramid. However, in this case, the height is the length of a line from the apex that makes a right angle with the base.

Let's take a look at some examples.

A square pyramid has a height of 9 meters. If the side of the base measures 4 meters, what is the volume of the pyramid?

Since the base is a square, area of the base = 4 * 4 = 16 m^2.

Volume of the pyramid = (1/3) * (base area) * (height)

V = (1/3) * (16) * (9) = 48 m^3

What is the surface area of the pyramid discussed in the first example?

In order to find the surface area, we must first find the slant height of the pyramid. Since we know the height and the base length, we can use the Pythagorean Theorem to find the slant length.

You can see that the blue and red lines create a right triangle. The length of the long leg of the triangle is 9 meters, or the height of the triangle. The length of the short leg of the triangle is 2 meters because it is half the length of the base of the triangle (4 m). The red line is the hypotenuse of the triangle and can be calculated using the formula:

*a*^2 + *b*^2 = *c*^2

2^2 + 9^2 = *c*^2

81 + 4 = *c*^2

85 = *c*^2

*c* = 9.2 m

Now that you know the slant height, you can solve for the surface area of this triangle.

SA = (base area) + (1/2) * (perimeter) * (slant height)

SA = 16 + (1/2) * (16) * (9.2) = 89.6 m^2

In geometry, a **pyramid** is a three-dimensional shape that can have any polygon as its base. The corners of the polygon all connect at the apex, or point, of the pyramid. There are formulas that can be used to determine the surface area and the volume of any pyramid. The formula for finding the surface area when all of the side faces are the same is:

SA = (base area) + (1/2) * (perimeter) * (slant height)

The **base area** is the area of the base and can be determined based on what figure the base is. For example, if the base of the pyramid is a square, the formula for finding the area is:

A = *s*^2

The **perimeter** is the distance around the base of the pyramid. The **slant height** is the diagonal height from the center of one of the base edges to the apex.

If the pyramid has side faces that differ from each other (like in the case of an irregular pyramid), then the surface area equation is:

SA = (base height) + (lateral area)

The volume of a pyramid can be found using this formula:

V = (1/3) * (base area) * (height)

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
13 in chapter 4 of the course:

Back To Course

High School Geometry: Homework Help Resource13 chapters | 142 lessons

- Area of Triangles and Rectangles 5:43
- Perimeter of Triangles and Rectangles 8:54
- How to Identify Similar Triangles 7:23
- Angles and Triangles: Practice Problems 7:43
- Triangles: Definition and Properties 4:30
- Classifying Triangles by Angles and Sides 5:44
- Interior and Exterior Angles of Triangles: Definition & Examples 5:25
- Constructing the Median of a Triangle 4:47
- Median, Altitude, and Angle Bisectors of a Triangle 4:50
- Constructing Triangles: Types of Geometric Construction 5:59
- Properties of Concurrent Lines in a Triangle 6:17
- Perfect Numbers: Definition, Formula & Examples 6:15
- Pyramid in Math: Definition & Practice Problems 5:31
- Transversal in Geometry: Definition & Angles 3:06
- What is a Hexagon? - Definition, Area & Angles 3:28
- What is a Right Angle? - Definition & Formula 3:19
- What is a Straight Angle? - Definition & Example 3:08
- What is an Obtuse Angle? - Definition & Examples 2:35
- What Is an Obtuse Triangle? - Definition & Area Formula 4:38
- Go to Properties of Triangles: Homework Help

- MTEL Sheltered English Immersion: Practice & Study Guide
- C (ASCP) Technologist in Chemistry: Study Guide & Exam Prep
- MLT (ASCP) Medical Laboratory Technician: Study Guide & Exam Prep
- Pennsylvania Grades 4-8 Core Assessment - Mathematics & Science (5155): Study Guide & Test Prep
- Literary Elements Lesson Plans & Resources
- Teaching Basic Reading Skills to English Language Learners
- Standards-Based Learning for MA ELL Students
- Instructional Strategies for Teaching Academic Language
- Second Language Acquisition in ELL Classrooms
- Encouraging Listening & Speaking Skills Development in ELL Classrooms
- California Code of Regulations for Schools
- WV Next Generation Standards for Math
- Continuing Education Opportunities for Microbiology Technologists
- Professional Publications in Literacy
- Dyslexia Programs in Texas
- Study.com's Teacher Edition
- Study.com School Plans

- Ocean Invertebrates: Sponges & Cnidarians
- Manures & Fertilizers: Types, Uses & Examples
- What Is a Service Business?
- How to Convert Meters per Second to Miles per Hour
- The Case of Lady Sannox: Summary, Characters & Setting
- The Bronze Age: Armor, Weapons & Warfare
- What is Crashing in Project Management? - Definition & Example
- Hemoglobin Testing: Purpose & Types
- Quiz & Worksheet - Characteristics of the Mod Subculture
- Quiz & Worksheet - Correlative Conjunctions
- Quiz & Worksheet - Rectification Definition
- Quiz & Worksheet - Lifelong Learning
- Quiz & Worksheet - Biological Assimilation of Food
- Graphs & Charts in Everyday Life Flashcards
- Interpreting & Analyzing Data Sets Flashcards

- Geology 101: Physical Geology
- DSST Environmental Science: Study Guide & Test Prep
- Religion 101: Intro to World Religions
- CSET Earth and Planetary Science Subtest III: Practice and Study Guide
- High School Algebra II: Help and Review
- The Criminal Justice Field: Help and Review
- Holt United States History Chapter 21: The Progressive Spirit of Reform (1868-1920)
- Quiz & Worksheet - Entropy
- Quiz & Worksheet - Features of Interpersonal Intelligence
- Quiz & Worksheet - Characteristics of Skin Cells
- Quiz & Worksheet - Landslides
- Quiz & Worksheet - Ammeter

- What Is Ecological Balance? - Definition & Importance
- Continuous Auditing: Definition, Techniques & Examples
- How Many Times Can You Take the TEAS Test?
- What Will I Learn in an SAT Class?
- What is a College Deferral Letter?
- 2nd Grade Common Core Math Standards
- The New SAT Math Section
- Student Loan Forgiveness in Texas
- Engineering Internships for High School Students
- Community College Teaching Jobs
- Adult Community Education
- Best Online High School English Courses

Browse by subject