Octagon
Calculate all properties of a regular octagon from side length: area, perimeter, all three diagonals, circumradius, and inradius. Free online geometry tool for students and professionals.
About This Calculator
The Octagon Calculator computes all key geometric properties of a regular octagon from a single side length input. Whether you are a student learning geometry, an architect designing an octagonal structure, or a DIY enthusiast planning octagon tiles or a gazebo, this tool gives you instant access to area, perimeter, all three diagonal lengths, circumradius, and inradius (apothem).
How It Works
A regular octagon is an eight-sided polygon with equal side lengths and equal interior angles of 135 deg. The calculator uses established geometric formulas to derive every property from the side length. The area is calculated using the formula A = 2(1 + sqrt2) x a^2, where a is the side length. The perimeter is simply eight times the side length. Three distinct diagonals are computed: the short diagonal connecting vertices with one vertex between them (s = a x sqrt(2 + sqrt2)), the medium diagonal connecting vertices with two vertices between them (m = a x (1 + sqrt2)), and the long diagonal connecting opposite vertices (l = a x sqrt(4 + 2sqrt2)). The circumradius R is half the long diagonal, and the inradius (apothem) r is half the medium diagonal.
What Is a Regular Octagon?
The word "octagon" comes from Greek roots: "octo" meaning eight and "gonia" meaning angle. A regular octagon has eight lines of symmetry and rotational symmetry of order 8. It can be constructed by truncating the corners of a square, which is why it appears frequently in architecture, tile patterns, and design. The sum of all interior angles is always 1080 deg, and each interior angle is 135 deg.
Common Uses
Octagons appear in many real-world applications: stop signs, camera aperture blades, architectural floor plans, roofing designs, tile patterns, gazebos, and decorative elements. Architects and builders use octagon geometry for pavilions, domes, and unique room layouts. In photography, the number of aperture blades (often 8) determines the shape of bokeh highlights. This calculator helps professionals and hobbyists quickly obtain accurate measurements without manual computation.
Frequently Asked Questions
What is a regular octagon?
A regular octagon is an eight-sided polygon where all sides and all interior angles are equal. Each interior angle measures 135 deg and the sum of all interior angles is 1080 deg.
How do you calculate the area of an octagon?
The area of a regular octagon with side length a is given by A = 2(1 + sqrt2) x a^2. This formula is derived by decomposing the octagon into a square and four right triangles, or by using the standard regular polygon area formula.
How many diagonals does an octagon have?
A regular octagon has 20 diagonals in total. There are three distinct lengths: short diagonals connecting vertices with one vertex between them, medium diagonals connecting vertices with two vertices between them, and long diagonals connecting opposite vertices.
What is the difference between circumradius and inradius of an octagon?
The circumradius (R) is the radius of the circumscribed circle that passes through all eight vertices of the octagon. The inradius (r), also called the apothem, is the radius of the inscribed circle that is tangent to all eight sides. For a regular octagon, the circumradius is always larger than the inradius.
What real-world objects are shaped like an octagon?
The most common real-world octagon is the stop sign. Octagonal shapes also appear in camera apertures, tiles and flooring patterns, architectural designs such as the Octagon House in Washington D.C., and some gazebos and pavilions.
Can this calculator work in any unit system?
Yes. Simply enter the side length in any unit (inches, feet, centimeters, meters, etc.) and all output values will be in the same unit squared for area and the same linear unit for perimeter, diagonals, and radii.
What is the formula for the diagonals of a regular octagon?
For side length a: short diagonal = a x sqrt(2 + sqrt2), medium diagonal = a x (1 + sqrt2), and long diagonal = a x sqrt(4 + 2sqrt2). The circumradius R equals half the long diagonal: R = a/2 x sqrt(4 + 2sqrt2).
Is the octagon calculator free to use?
Yes, all calculators on Calculy including this octagon calculator are completely free to use with no registration or download required.