Octagon: Properties, Angles, and Area Formula

TL;DR

An octagon is an eight-sided polygon whose interior angles sum to 1080°; in a regular octagon, each interior angle is 135° and each exterior angle is 45°. This article defines the octagon, derives its area formula A=2(1+√2)s², counts its 20 diagonals, and works through examples — starting from the stop sign in every intersection.

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:

Sum of interior angles=(n−2)×180°=(8−2)×180°=6×180°=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:

Each interior angle=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 n(n−3)/2 with n=8: 8×5/2=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.

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 1/2 s a, so the whole octagon is:

A=8×1/2 s a=1/2 (8s) a=1/2 P a

where P=8s is the perimeter. This A=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=s/2(1+√2). Substituting gives the side-only formula:

A=2(1+√2)s²≈4.828s²

In this formula s is the side length, the factor 2(1+√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)×180°=6×180°=1080°

Each angle of a regular octagon:

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 1080°/6=180°. Spot the error.

A natural first move is to divide the angle sum by the number of angles instead, which is 8:

1080°/8=135°

Each interior angle is 135°.

Example 3

A regular octagon has a side length of 5 cm. Find its perimeter and area.

Perimeter is eight equal sides:

P=8×5=40 cm

Area uses the side-only formula:

A=2(1+√2)(5)²=2(2.414)(25)≈120.7 cm²

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:

n(n−3)/2=20.

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.

Area:

A=1/2 P a=1/2×40×6=120 cm²

Example 6

A tiler lays regular octagonal tiles and fills the gaps with small squares. Show why the tiles fit with no gaps.

A regular octagon's interior angle is 135°. At each meeting point, two octagon corners contribute 135°+135°=270°, leaving 90° for the square.

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+√2)s², derived from A=1/2 P a by splitting the octagon into 8 triangles.
• A regular octagon is convex, with eight lines of symmetry.