Back To Course

CAHSEE Math Exam: Help and Review22 chapters | 255 lessons | 13 flashcard sets

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:
*David Karsner*

In geometry, the octahedron is one of five Platonic solids. It is formed from eight equilateral triangles. If you took two congruent square pyramids and connected their bases, you would create an octahedron.

The **octahedron** is one of five solids in the set of Platonic solids, and the one with eight faces. **Platonic solids** get their name from Plato, not because he discovered them but because he mentioned them often in his writings. He assigned each solid to be a representation of the natural elements. The octahedron was the symbol for air. The other four Platonic solids are the tetrahedron, cube, dodecahedron, and the icosahedron.

The octahedron is a three-dimensional object and is composed of one- and two-dimensional parts. These parts have special names.

The base of an octahedron is a square. If you picture an octahedron as two congruent square pyramids that have their bottoms touching, then the base of the octahedron is the square between the two pyramids.

An octahedron has eight faces, which are all in the shape of equilateral triangles. These eight faces are where the solid gets its name. 'Octa' means eight. These faces form the surface area of the octahedron. The square that is the base of the octahedron is not part of the surface area; therefore, the base isn't also a face.

When two faces touch, the line segment that is formed is called an edge. An octahedron has 12 edges.

When two edges intersect they form a vertex (plural being vertices). The octahedron has six vertices. Each vertex is formed when four edges intersect.

Notice how each vertex has four edges that touch it. Those edges connect the vertex with four of the other five vertices. There is one vertex called the non-adjacent vertex that isn't connected to the vertex by an edge. The distance from any vertex to its non-adjacent vertex will always be the length of any edge times the square root of two (1.414). The line formed by creating the vertex to non-adjacent vertex is the hypotenuse of a right triangle. Two of the edges form the legs of the right triangle.

Since a right triangle has been created, you can use the Pythagorean Theorem *a*^2 + *b*^2 = *c*^2 where *a* and *b* are legs and *c* is the hypotenuse to find the distance between any vertex and its non-adjacent vertex. The edge is *x* in length. *x*^2 + *x*^2 = *c*^2, which simplifies to 2*x*^2 = *c*^2. Square root both sides and you have *c*=*x*(square root of 2).

Since the surface of an octahedron consists of eight congruent triangles, the surface area of an octahedron would be the area of one of the triangles times eight. The formula is Surface Area=4*bh*, where *b* is the length of the base of the triangle and *h* is the length of the height of the triangle.

To find the surface area of an octahedron, you'll need to know the length of the base of the triangle and the height.

In this octahedron, the base is 12 inches and the height is 10.4 inches. The formula for surface area is 4*bh*. If we substitute 12 for *b* and 10.4 for *h* we have S.A. = 4(12)(10.4) = 499.2. Area is always measured in square units, so the surface area of this octahedron is 499.2 square inches.

An octahedron consists of two square pyramids. To find the volume of an octahedron, find the volume of one the pyramids and multiply by two. The volume of a pyramid is *b*^2(*h*)/3, where *b* is the length of one side of the square that forms the base and *h* is the height of the pyramid. The formula for the volume of an octahedron would be 2*b*^2(*h*) / 3.

To find the volume of an octahedron, you need to know the length of the square base and the height of the pyramid. In this example, the length of the square base is 12 inches (the same as the base of the triangle) and the height of the pyramid is 8.5 inches. The formula for volume of an octahedron is *V*=2*b*^2(*h*)/3. Therefore the volume of the octahedron is 2(144)(8.5) / 3 = 816. Volume is measured in cubic units, so the volume of this octahedron is 816 cubic inches.

Let's review. **The octahedron** is one of five solids in the set of Platonic solids, and the one with eight faces. **The Platonic solids** are solid shapes that the philosopher Plato assigned each to be a representation of the natural elements. The octahedron is created by connecting two square pyramids. It has eight faces, six vertices, and twelve edges. The surface area can be obtained using the formula Surface Area = 4*b**h*. The formula for the volume is *V* = *b*^2(*h*) / 3.

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 16 of the course:

Back To Course

CAHSEE Math Exam: Help and Review22 chapters | 255 lessons | 13 flashcard sets

- Properties of Shapes: Rectangles, Squares and Rhombuses 5:46
- Properties of Shapes: Quadrilaterals, Parallelograms, Trapezoids, Polygons 6:42
- Properties of Shapes: Triangles 5:09
- Properties of Shapes: Circles 4:45
- What are 3D Shapes? - Definition & Examples 3:28
- Octahedron: Definition & Properties 5:08
- Rectangle: Types, Properties & Formulas 4:52
- Tessellation: Definition & Examples
- Go to CAHSEE - Properties of Shapes: Help and Review

- Computer Science 336: Network Forensics
- Computer Science 220: Fundamentals of Routing and Switching
- Global Competency Fundamentals & Applications
- Introduction to the Principles of Project Management
- Praxis Elementary Education: Reading & Language Arts - Applied CKT (7902): Study Guide & Practice
- Practical Applications for Business Ethics
- Practical Applications for Marketing
- Practical Applications for HR Management
- Practical Applications for Organizational Behavior
- Analyzing Texts Using Writing Structures
- TASC Test Score Information
- What is the TASC Test?
- Praxis Prep Product Comparison
- GED Prep Product Comparison
- CBEST/CSET Prep Product Comparison
- ASVAB Prep Product Comparison
- GACE Prep Product Comparison

- Developing & Managing a High-Quality Library Collection
- Library Space Planning
- Literacy Strategies for Teachers
- Modeling Oral & Written Communication Skills in the Classroom
- Practical Application: Understanding Employee Behavior
- Positive Global Outcomes of Global Competence
- Practical Application: Color Wheel Infographic
- Practical Application: Making Quantitative Decisions in Management
- Quiz & Worksheet - Developing a Learner-Centered Classroom
- Quiz & Worksheet - Pectoralis Major Anatomy
- Quiz & Worksheet - Technology for Teaching Reading
- Quiz & Worksheet - Professional Development Ideas for Teachers
- Quiz & Worksheet - How to Teach Reading to ELL Students
- Flashcards - Measurement & Experimental Design
- Flashcards - Stars & Celestial Bodies

- Ancient Egypt Study Guide
- Calculus for Teachers: Professional Development
- Praxis Spanish Exam (5195): Practice & Study Guide
- Accuplacer ESL Sentence Meaning Test: Practice & Study Guide
- Accuplacer Math: Quantitative Reasoning, Algebra, and Statistics Placement Test Study Guide
- The Beginning of World War II: Tutoring Solution
- PSAT Math - Geometry and Measurement: Help and Review
- Quiz & Worksheet - Life in Different Locations in Australia & the Pacific Islands
- Quiz & Worksheet - Roles for Effective Team Collaboration
- Quiz & Worksheet - Logos in Writing
- Quiz & Worksheet - Climax in Literature
- Quiz & Worksheet - Oxymorons in Literature

- Jupiter, Roman God: Facts & Myths
- Days Sales Outstanding (DSO): Definition & Formula
- AP English Literature Question Types
- English Language Learning Programs in California
- Cause & Effect Experiments for Kids
- MCAT Tips
- Static Electricity Experiments for Kids
- What is the Center for Change in Utah?
- How to Ace a Group Interview
- What's in Common Core Standards Appendix C?
- How to Become a User Interface Designer
- Science Projects for Preschoolers

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

Browse by subject