A stop sign needs no text to be understood — its eight-sided shape is recognised even from behind, in fog, or under snow. That is an octagon doing a job no other polygon was chosen for.
An octagon is a polygon with eight straight sides and eight vertices ("octa" means eight). A regular octagon has all eight sides equal and all eight angles equal; an irregular octagon has eight sides of differing lengths or angles. Like every polygon, its interior angles follow the sum rule, which for eight sides gives $1080°$. The octagon sits beside the pentagon and hexagon as one of the most common named polygons; for the family it belongs to, see polygons.
By the end you will know why a regular octagon's angles are $135°$, how many diagonals it has, and where its area formula comes from. The center-to-side distance marked above is the apothem, and it is the key to the octagon's area.
Angles of an Octagon
Every octagon, regular or not, has interior angles that sum to the same total. Use the polygon angle-sum formula with $n = 8$:
$$\text{Sum of interior angles} = (n-2)\times 180° = (8-2)\times 180° = 6 \times 180° = 1080°$$
Here $n = 8$ is the side count and $(n-2) = 6$ is the number of triangles the diagonals from one vertex carve the octagon into. For a regular octagon, the eight equal angles share that total:
$$\text{Each interior angle} = \frac{1080°}{8} = 135°$$
The exterior angle at each vertex is the supplement, $180° - 135° = 45°$, and the eight exterior angles sum to $360°$, as they do for every polygon. These are the same interior and exterior angles defined in interior angles.
Properties of a Regular Octagon
A regular octagon's symmetry gives it a clean set of properties worth knowing before any calculation.
8 sides, 8 vertices, 8 lines of symmetry. It maps onto itself under rotations of $45°$.
Interior angle $135°$, exterior angle $45°$. The interior-angle sum is $1080°$.
20 diagonals. Using $\dfrac{n(n-3)}{2}$ with $n = 8$: $\dfrac{8 \times 5}{2} = 20$.
It is convex. Every interior angle ($135°$) is below $180°$, so a regular octagon is a convex polygon.
It can be split into 8 equal isosceles triangles from the centre — the fact that powers the area formula below.
Deriving the Area of a Regular Octagon
Rather than memorise the area formula, build it from the apothem — the perpendicular distance from the centre to the middle of a side.
Slice the regular octagon from its centre to every vertex. This produces 8 identical isosceles triangles, each with base $s$ (a side of the octagon) and height $a$ (the apothem). The area of one triangle is $\tfrac{1}{2},s,a$, so the whole octagon is:
$$A = 8 \times \tfrac{1}{2},s,a = \tfrac{1}{2},(8s),a = \tfrac{1}{2},P,a$$
where $P = 8s$ is the perimeter. This $A = \tfrac{1}{2},P,a$ is the universal area formula for any regular polygon, derived in full from the apothem above. For the octagon, the apothem in terms of the side is $a = \tfrac{s}{2}(1+\sqrt{2})$. Substituting gives the side-only formula:
$$A = 2(1+\sqrt{2}),s^2 \approx 4.828,s^2$$
In this formula $s$ is the side length, the factor $2(1+\sqrt{2})$ is a fixed constant for every regular octagon, and the result is in square units of whatever unit $s$ uses.
Examples of Octagon
Example 1
Find the sum of the interior angles of an octagon and each angle of a regular octagon.
Sum of interior angles:
$$(8-2)\times 180° = 6 \times 180° = 1080°$$
Each angle of a regular octagon:
$$\frac{1080°}{8} = 135°$$
The interior angles total $1080°$, and each regular-octagon angle is $135°$.
Example 2
A student computes a regular octagon's interior angle as $\dfrac{1080°}{6} = 180°$. Spot the error.
A natural first move is to divide the angle sum by the number of triangles, $6$. Try it: $1080° \div 6 = 180°$. But a $180°$ "angle" is a straight line, which a real corner cannot be, and that signals the divisor is wrong.
Divide by the number of angles instead, which is $8$:
$$\frac{1080°}{8} = 135°$$
Each interior angle is $135°$. The $(n-2)$ counts triangles; the divisor for each angle is always $n$.
Example 3
A regular octagon has a side length of $5$ cm. Find its perimeter and area.
Perimeter is eight equal sides:
$$P = 8 \times 5 = 40 \text{ cm}$$
Area uses the side-only formula:
$$A = 2(1+\sqrt{2}),s^2 = 2(1+1.414)(5)^2 = 2(2.414)(25) \approx 120.7 \text{ cm}^2$$
The perimeter is $40$ cm and the area is about $120.7$ cm².
Example 4
How many diagonals does an octagon have?
Use the diagonal formula for an $n$-gon:
$$\frac{n(n-3)}{2} = \frac{8(8-3)}{2} = \frac{8 \times 5}{2} = \frac{40}{2} = 20$$
An octagon has 20 diagonals.
Example 5
A regular octagon has an apothem of $6$ cm and a side of $5$ cm. Find its area using the perimeter-apothem formula.
Perimeter:
$$P = 8 \times 5 = 40 \text{ cm}$$
Area:
$$A = \tfrac{1}{2},P,a = \tfrac{1}{2}\times 40 \times 6 = 120 \text{ cm}^2$$
The area is $120$ cm². This matches the side-only formula closely, confirming the two routes agree.
Example 6
A tiler lays regular octagonal tiles and fills the gaps with small squares. Show why the tiles fit with no gaps, and find the square's corner angle.
A regular octagon's interior angle is $135°$.
At each meeting point the angles around it must total $360°$.
Two octagon corners contribute $135° + 135° = 270°$, leaving $360° - 270° = 90°$ for the square.
A $90°$ corner is exactly a square's angle, so two octagons and one square close the full turn with no gap. This octagon-and-square pattern is one of the classic floor tilings, and it works because of the $135°$ octagon angle.
Where the Octagon Earns its Keep: Visibility by Shape Alone
The octagon was chosen for the stop sign for a reason that is pure geometry, and the same logic recurs wherever a shape must be read instantly.
The stop sign. In 1922 American highway engineers gave the stop sign eight sides so it could be told apart from every other sign by shape alone — even from the back, where a driver sees the outline but not the word. More sides than a triangle or square meant "more important," and eight was the agreed standard.
Architecture and design. Octagonal floor plans — the Dome of the Rock, countless gazebos and bay windows — give nearly the openness of a circle while still being buildable from straight walls and standard materials.
Why eight sides specifically. The destination is recognisability with buildability. An octagon is round enough to stand out yet still made of straight, cuttable edges, hitting the sweet spot a circle (no straight sides) and a square (too ordinary) both miss.
The eight-sided standard for the stop sign was set by the Mississippi Valley Association of State Highway Departments and spread worldwide — a geometry decision now obeyed by every driver on the planet.
Octagon Mistakes to Watch For
Mistake 1: Dividing the angle sum by $6$ instead of $8$
Where it slips in: Finding each interior angle of a regular octagon right after computing the $1080°$ sum.
Don't do this: Write $\dfrac{1080°}{6} = 180°$, confusing the triangle count $(n-2)$ with the angle count $n$.
The correct way: The sum $1080°$ is shared among the eight angles, so divide by $8$ to get $135°$. The student who blurs the two numbers gets a meaningless $180°$ "corner" and should treat that absurd result as a flag to recheck.
Mistake 2: Using the regular area formula on an irregular octagon
Where it slips in: Applying $A = 2(1+\sqrt{2}),s^2$ to an octagon whose sides are not all equal.
Don't do this: Plug one side length into the regular formula when the eight sides differ.
The correct way: The formula $2(1+\sqrt{2}),s^2$, and the apothem itself, exist only for regular octagons. For an irregular one, split it into triangles and add their areas. The memoriser who learned "octagon area = $2(1+\sqrt{2})s^2$" as a fact applies it to the first lopsided eight-sided plot and gets the wrong acreage.
Mistake 3: Forgetting units when reporting area
Where it slips in: Writing the area as a bare number with no square units.
The correct way: Area is always in square units. If the side is in centimetres, the area is in cm². Carry the unit through every line, or omit units entirely and state so — never mix.
Key Takeaways
An octagon has 8 sides; its interior angles sum to $1080°$.
A regular octagon has each interior angle $135°$, each exterior angle $45°$, and 20 diagonals.
The area is $A = 2(1+\sqrt{2}),s^2$, derived from $A = \tfrac{1}{2},P,a$ by splitting the octagon into 8 triangles.
The regular formula applies only when all eight sides are equal; irregular octagons are split into triangles instead.
A regular octagon is convex, with eight lines of symmetry.
A Practical Next Step
Practice these problems to solidify your understanding. Sketch each octagon and label its angle, side, and apothem before calculating.
A regular octagon has a side of $4$ cm. Find its area. (Answer to Question 1: $2(1+\sqrt{2})(4)^2 \approx 77.3$ cm².)
Find each exterior angle of a regular octagon. (Answer to Question 2: $360° \div 8 = 45°$.)
To work through octagons and other polygons with a teacher, explore Bhanzu's geometry tutor, our middle school math tutor sessions, or math classes online. To see a trainer derive the octagon area live, you can book a free demo class.
Read More
Types of polygon — where the regular octagon sits among all polygon classes.
Convex polygon — why the regular octagon's $135°$ angles make it convex.
Exterior angles of a polygon — the $45°$ octagon exterior angle as part of the $360°$ rule.
Angles in a pentagon — the same interior-angle reasoning for five sides.
Geometric shapes — the wider 2-D and 3-D shape family.
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