What Is An Eight-Sided Shape? | Octagon Facts That Stick

An eight-sided polygon is called an octagon, a shape with eight straight sides and eight corners.

You searched What Is An Eight-Sided Shape? because you want the real name, plus a clear way to spot it and work with it in math class. Good news: the core idea is simple, and the useful details are easy once you know where to look.

An eight-sided shape can show up as a perfect “stop-sign style” figure, or as a wonky outline you’d sketch on graph paper. Both count, as long as the boundary is made of straight line segments and there are eight of them.

What An Eight-Sided Shape Is In Geometry Class

An octagon is a polygon with eight sides. That means the outline is built from eight straight segments that connect end to end, forming a closed loop. Where two segments meet, you get a corner (a vertex). In an octagon, there are eight vertices.

If the “eight-sided shape” you’re looking at has curved edges, it isn’t a polygon, so it isn’t an octagon in the geometry sense. Curves can still be cool shapes, just not polygons.

One clean way to check quickly: trace the boundary with your finger and count the straight segments. If you can count eight straight edges without lifting your finger, you’ve got an octagon.

Octagon Vs Regular Octagon

People often say “octagon” and picture a neat, symmetric shape. That neat version has a special name: a regular octagon.

Irregular octagon

An irregular octagon still has eight sides, yet the side lengths can differ and the angles can differ. It can look lopsided, stretched, or skewed. It still counts as long as it has eight straight sides.

Regular octagon

A regular octagon has:

• Eight equal sides
• Eight equal interior angles

This is the version that behaves nicely in formulas, symmetry, tiling puzzles, and “find the angle” exercises.

Parts Of An Octagon You Should Know

When teachers talk about an octagon, they usually mean these parts. Once you can name them, worksheet questions stop feeling cryptic.

Sides and vertices

Each straight segment is a side. Each corner is a vertex. An octagon has eight sides and eight vertices, always.

Interior angles

An interior angle is the angle you see inside the shape at a vertex. For a regular octagon, every interior angle has the same measure.

Diagonals

A diagonal connects two non-adjacent vertices. Diagonals help with counting triangles, building area methods, and spotting symmetry.

Perimeter and area

Perimeter is the total distance around the outside. Area is the amount of flat space inside the boundary.

Angle Facts That Make Octagon Problems Easier

Angle questions are where many students trip. The trick is to separate two ideas: the sum of interior angles, and the measure of each interior angle in a regular octagon.

Interior angle sum for any octagon

For any polygon with n sides, the sum of interior angles is:

(n − 2) × 180°

For an octagon, n = 8, so the sum is:

(8 − 2) × 180° = 6 × 180° = 1080°

Each interior angle in a regular octagon

If the octagon is regular, all eight interior angles match. So each one is:

1080° ÷ 8 = 135°

Exterior angles (the quick loop around)

Exterior angles are the “turning angles” you’d make if you walked around the shape. For any convex polygon, the exterior angles add to 360°.

So for a regular octagon, each exterior angle is:

360° ÷ 8 = 45°

Eight-Sided Shape Name And Quick Properties

This is the short list students wish they had on page one. It’s the same shape, just viewed through different “what does the question want?” lenses.

If you want a dictionary-style definition for the name, Britannica’s entry matches the classroom meaning of an octagon as a flat shape with eight sides and angles: Britannica’s octagon definition.

If you want a math reference that states the octagon definition and connects it to the regular case, Wolfram’s page is a strong checkpoint: Wolfram MathWorld’s Octagon page.

Octagon Formulas You’ll Use Most

Formulas depend on what you’re given. Teachers mix the givens on purpose, so it helps to keep a menu in your head.

Perimeter

Any octagon: add all side lengths.

Regular octagon: P = 8s, where s is one side length.

Area of a regular octagon using side length

A common closed-form area formula for a regular octagon with side length s is:

A = 2(1 + √2)s²

Area using apothem (regular octagon)

The apothem is the distance from the center to the midpoint of a side (for a regular polygon). With apothem a and perimeter P:

A = (1/2)Pa

This is often the smoothest route on tests because it reduces the job to “find P” and “find a,” then multiply.

Diagonal count

The number of diagonals in an n-gon is:

n(n − 3) ÷ 2

For an octagon: 8(8 − 3) ÷ 2 = 8 × 5 ÷ 2 = 20 diagonals.

That diagonal count shows up in combinatorics-style questions, and it’s a tidy check for your understanding.

Octagon Facts At A Glance

This table bundles the facts that pop up most in homework, quizzes, and geometry proofs. Use it as a “scan and go” section when you get stuck mid-problem.

Feature	Regular Octagon	Notes
Number of sides	8	Must be straight segments
Number of vertices	8	One vertex per corner
Interior angle sum	1080°	True for any octagon
Each interior angle	135°	Only when regular
Each exterior angle	45°	Only when regular (convex)
Number of diagonals	20	n(n − 3) ÷ 2 with n = 8
Lines of symmetry	8	4 through opposite vertices, 4 through side midpoints
Rotational symmetry	Order 8	Turns of 45° map it to itself
Perimeter	P = 8s	s is one side length
Area (side length)	A = 2(1 + √2)s²	Regular case formula
Area (apothem)	A = (1/2)Pa	P is perimeter, a is apothem

How To Spot An Octagon Fast

On worksheets, the tricky part is that drawings are not always neat. Some are rotated, stretched, or drawn without equal sides.

Count sides, not “look”

Start with counting straight sides. If it has eight, it’s an octagon even if it looks odd.

Check for straight edges

A shape with eight “bumps” that are curved is not an octagon. A polygon’s edges are straight. That single rule knocks out a lot of traps.

Watch for shared edges in composite figures

Sometimes a picture has many segments and it’s easy to count wrong. Trace only the outer boundary. Ignore interior lines unless the question tells you they matter.

Ways To Find Area When The Octagon Is Not Regular

Most real drawings are irregular. That doesn’t block you. It just changes your plan. Here are methods that work well in classwork and exams.

Split it into triangles

Pick one vertex, draw diagonals to non-adjacent vertices, and you can split an octagon into triangles. Then add triangle areas. This method is common because triangle formulas are familiar.

Use rectangles and right triangles on grid paper

If the octagon sits on a grid, you can often box it into a rectangle, then subtract corner triangles. The shape looks messy at first, then it turns into small, clean chunks.

Coordinate geometry method

If you have vertex coordinates, you can compute area with a coordinate area routine (often taught as the “shoelace” method). This can feel mechanical, yet it’s reliable when the vertices are listed in order.

Turn it into a regular-octagon problem when you can

Sometimes the drawing is regular and the worksheet gives a side length, apothem, or radius from the center. In that case, use the regular polygon formulas from earlier and save time.

Area And Perimeter Methods Compared

This table helps you pick a method based on the information you have in the problem statement.

Method	What you need	When it helps
Perimeter sum	All side lengths	Any octagon, regular or not
Regular perimeter	One side length (s)	Fast when all sides match
Regular area (side length)	One side length (s)	Algebra-friendly problems
Regular area (apothem)	Perimeter (P) and apothem (a)	Center-based diagrams
Triangle split	Ways to find triangle areas	Irregular shapes with clear diagonals
Box-and-subtract	Bounding rectangle and corner pieces	Grid drawings with right angles
Coordinate area routine	All vertex coordinates in order	Analytic geometry tasks

How To Draw A Clean Regular Octagon By Hand

If you can draw one cleanly, you can label it, measure it, and avoid silly mistakes in angle work. Here are two classroom-friendly methods.

Method 1: Start from a square

1. Draw a square.
2. Mark equal distances from each corner along both sides meeting at that corner.
3. Connect those marks with straight lines to “cut off” each corner.
4. You’ll get an eight-sided figure. If the cut lengths match, it will be regular.

This is a go-to method for sketches because it builds symmetry right away.

Method 2: Use a circle and compass marks

1. Draw a circle with a compass.
2. Draw a horizontal diameter and a vertical diameter through the center.
3. Bisect each right angle at the center to make 45° directions.
4. Mark the eight points where those rays meet the circle.
5. Connect the points in order with straight segments.

This makes vertices equally spaced around the circle, which produces a regular octagon.

Common Mistakes And How To Dodge Them

These are the errors that keep showing up in student work, even when the student understands the topic.

Mixing up “sides” and “angles”

An octagon has eight sides and eight angles. That’s always true. If your count gives eight sides and seven corners, something went wrong in the tracing.

Using the regular angle value on an irregular octagon

135° is tied to a regular octagon. If the sides are not equal, the angles do not have to match. Use 1080° as the total sum, then work from the given clues.

Counting diagonals by drawing and guessing

Drawing all diagonals gets messy fast. Use the formula n(n − 3) ÷ 2 with n = 8 to get 20, then move on.

Perimeter errors from double-counting

In composite figures, students sometimes add interior segments that are not part of the boundary. Trace the outside edge only. If your pencil crosses an interior line, stop and restart the trace.

Mini Practice Set You Can Do On Paper

Try these to lock the ideas in. No tricks, just the kind of tasks that match typical classwork.

Practice 1: Interior angle sum

Write the interior angle sum of an octagon. Then write the measure of each interior angle if it is regular.

Practice 2: Perimeter

A regular octagon has side length 6 cm. Find its perimeter.

Practice 3: Exterior angles

A regular octagon’s exterior angles are all equal. Find the measure of one exterior angle, then state how many such turns complete a full rotation.

Practice 4: Spot the octagon

Sketch three different eight-sided polygons: one regular, one convex but irregular, and one that has an inward “dent” (concave). Label the eight vertices on each sketch to prove your count.

One-Page Octagon Checklist

Use this as a final pass before you submit a worksheet or test answer.

• Did I count eight straight sides on the outer boundary?
• Did I keep 1080° as the interior angle sum for any octagon?
• Did I use 135° only when the octagon is regular?
• Did I use P = 8s only when all sides match?
• Did I choose an area method that fits the given info (side length, apothem, grid, or coordinates)?
• Did I avoid adding interior lines into the perimeter?