Octagon in Geometry: Definition, Properties & Formula

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  • 0:05 What is an Octagon?
  • 0:41 Properties of an Octagon
  • 1:57 Area of an Octagon
  • 2:42 Lesson Summary
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Lesson Transcript
Instructor: Cathryn Jackson

Cat has taught a variety of subjects, including communications, mathematics, and technology. Cat has a master's degree in education and is currently working on her Ph.D.

Octagons appear in real life in all sorts of places: architecture, engineering, and even nature. In this lesson, explore the properties of an octagon, learn how to find the area of a regular octagon and review what you've learned with a short quiz.

What Is an Octagon?

An octagon is a geometrical shape that contains eight sides and eight angles. Octagons must have straight sides that connect; they cannot be curved or disconnected. You will often see octagons in real life in the shape of a stop sign.

Octagon Shape in Stop Sign
Octagon Shape in Stop Sign

At one time, you could also see octagons used in architecture. Houses were once made in North America in an octagonal shape because they were believed to be cheaper to build and easier to heat and cool. Few of these houses remain standing today, but we can still appreciate the usefulness of octagons in other areas of life!

Properties of an Octagon

Octagons can either be regular or irregular. Any shape with eight straight, connected sides is an octagon. Where and how those sides connect, and the length of the sides, determines whether or not the octagon is regular.

Regular octagons have sides that are congruent. That means that all of the sides of the octagon are the same measurement. They also have congruent angles, which means that the measurement of all of the angles is the same, too. The exterior angles of a regular octagon are all 135 degrees and the interior angles are all 45 degrees.

Interior and Exterior Angles of a Regular Octagon
Interior and Exterior Angles of a Regular Octagon

As you can see from the image above, exterior angles are the angles along the edge of the octagon, while interior angles are created by drawing two diagonals inside the octagon.

Irregular octagons have sides that are not congruent, which means that they aren't all the same measurement. They can also have points that are facing inward and outward. An octagon that has points that are facing inward is a concave octagon.

Irregular Octagons
Irregular Octagons

It is important to note that regular octagons are always classified as convex octagons, while irregular octagons can be either concave or convex.

Area of an Octagon

The area of a regular octagon can be found by using this formula:

Area of an Octagon Formula
Area of an Octagon Formula

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