# 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](/content/math/geometry/pentagon-shape/index.html) and [hexagon](/content/math/geometry/hexagon/index.html) as one of the most common named polygons; for the family it belongs to, see [polygons](/content/math/geometry/polygons/index.html). 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](/content/math/geometry/apothem/index.html), 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](/content/math/geometry/interior-angles/index.html). ## 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.