Free Octagon Calculator

About the Octagon Calculator

The regular octagon calculator is a free online geometry tool that quickly computes the key properties of a regular octagon: area, perimeter, diagonal lengths (short, medium, and long), circumradius, and inradius. You only need to supply one value—most often the side length—and the calculator returns all other dimensions automatically. This makes it equally useful as a Regular Octagon Area Calculator, Octagon Perimeter Calculator, and Octagon Diagonal Length Calculator. Whether you are planning a tiling layout, designing a structural element, or studying geometry, this tool provides reliable results without requiring manual formula work.

What Is a Regular Octagon?

An octagon is a polygon with eight sides and eight vertices. The word comes from Greek roots meaning “eight angles.” In a regular octagon, all sides have the same length and all interior angles are equal. Each interior angle measures $135^\circ$, and the sum of the eight interior angles of any octagon is $1080^\circ$ (following the general polygon rule $(n-2)\times180^\circ$ for $n=8$). This uniformity makes the regular octagon a favorite shape in architecture, design, and mathematics.

Area of a Regular Octagon

The area can be calculated using a straightforward formula that depends only on the side length $a$:

$$A = 2(1+\sqrt{2})\,a^{2}$$

For example, if the side length is $5\ \text{cm}$, the area becomes $2(1+\sqrt{2})\times5^{2} \approx 120.71\ \text{cm}^{2}$. Alternatively, you can use the generic regular‑polygon formula: $A = \frac{P \times a_p}{2}$, where $P = 8a$ (the perimeter) and $a_p = \frac{a}{2}(1+\sqrt{2})$ (the apothem). Both approaches give the same result and are supported by this calculator.

Alternative Decomposition Methods

One intuitive way to derive the area is to view the octagon as a large square with four right‑angle triangles cut from its corners. If the original square has side $2a_p$, the removed triangles have legs equal to $\frac{a}{\sqrt{2}}$. Subtracting the four triangles from the square yields the octagon’s area. Another decomposition splits the octagon into a central rectangle and two identical trapezoids. These methods reinforce the geometry behind the formula and are helpful when working without a calculator.

Diagonals of a Regular Octagon

A regular octagon has 20 diagonals in total, falling into three length categories:

• Short diagonal ($s$), connecting vertices that are two steps apart:

  $$s = a\sqrt{2+\sqrt{2}}$$

• Medium diagonal ($m$), also called the height, connecting vertices that are three steps apart:

  $$m = a(1+\sqrt{2})$$

• Long diagonal ($l$), connecting opposite vertices through the center:

  $$l = a\sqrt{4+2\sqrt{2}}$$

These formulas allow you to quickly determine the necessary spacing when designing frames, brackets, or any structure that involves regular octagons.

Circumradius and Inradius

The circumradius ($R$) is the radius of the circumscribed circle that passes through all eight vertices, and it equals half the long diagonal:

$$R = \frac{l}{2} = \frac{a}{2}\sqrt{4+2\sqrt{2}}$$

The inradius ($r$), or apothem, is the radius of the inscribed circle tangent to each side:

$$r = \frac{m}{2} = \frac{a}{2}(1+\sqrt{2})$$

Both radii can be read directly from the calculator, simplifying tasks such as fitting an octagon inside a given circular boundary or encasing it within a circular frame.

How to Draw a Regular Octagon

Two practical methods are commonly used:

1. Circle method: Draw a circle and mark eight points on its circumference that are $45^\circ$ apart (obtained by dividing the full $360^\circ$ by 8). Connect these points with straight lines to form a regular octagon.

2. Square‑truncation method: Start with a square of side $a(1+\sqrt{2})$. Cut off each corner along a $45^\circ$ line so that the remaining side length is $a$. This produces a regular octagon without needing a compass or protractor.

Both techniques are easy to carry out with basic drawing tools and yield a precise regular octagon.

Using the Octagon Calculator

The calculator interface presents a single input field for the side length. Once you enter a value, it instantly populates all other fields: perimeter, area, short/medium/long diagonals, circumradius, and inradius. No additional clicks or conversions are needed. This simplicity makes the tool an ideal Geometry Calculator for anyone who needs quick, accurate results without performing multi‑step manual calculations.

Real‑World Octagonal Shapes

The regular octagon appears more often than one might think. The famous Octagon House in Washington, D.C., is an eight‑sided residential building. Octagonal floor tiles are popular because they can be combined with square tiles to cover a floor completely. In photography, camera apertures often adopt an octagonal shape; the number of aperture blades matches the polygon’s sides, producing an eight‑pointed star pattern in out‑of‑focus highlights. These are just a few examples that show how the properties of a regular octagon translate into practical design and everyday objects.