Octagon Calculator - Area, Apothem, and Circumradius

An octagon calculator is a geometry tool that finds the area, apothem, perimeter, and circumradius of a regular octagon from a single known measurement. It is most useful for students, designers, and engineers who need reliable octagon numbers without redoing the trig every time. Enter the side length, apothem, or area and the tool returns the full set of values in the same units.

• • Geometry homework: Solve regular octagon word problems when only one measurement is given and the rest need to be derived.
• • Stop sign and signage design: Estimate the area of a stop sign or other octagonal sign panel when only the side length is given on a drawing.
• • Tessellation and tile planning: Estimate the area an octagon will cover when it is part of a repeating tile or floor plan with eight equal sides.
• • Puzzle and craft patterns: Calculate fabric, paper, or material needed for octagon-based crafts without manual trig.

Regular octagons are everywhere in everyday geometry. The stop sign on a US street, the shape of a common umbrella, and many quilt blocks are all regular octagons. Because the shape is built from eight equal sides and eight equal interior angles of 135 degrees, every measurement can be derived from one side length, which is what makes the calculator useful.

Use the octagon calculator any time an octagon is described as regular, equilateral, or with eight equal sides. If the shape is irregular, the formulas on this page no longer apply and a vertex-by-vertex method is required.

When the shape you are working on is not a regular octagon, Polygon Area Calculator lets you supply vertex coordinates and still get a clean area result.

The calculator uses the regular-polygon area formula specialized to eight sides. Once the side length is known, the same value powers the perimeter, apothem, and circumradius.

• s: Side length, the length of one of the eight equal sides.
• A: Area of the octagon in the same square units as s.
• a: Apothem, distance from the center to the midpoint of a side.
• P: Perimeter, equal to 8s.
• R: Circumradius, distance from the center to a vertex.

The formula is the general regular-polygon area formula A = (n/4) * s^2 * cot(pi/n) with n = 8. The cot(pi/8) constant equals 1 + sqrt(2) and is approximately 2.4142135624, so the area factor collapses to about 4.8284 for every unit of side length.

When the user types an apothem, the calculator inverts the relationship a = s * (1 + sqrt(2)) / 2 to recover the side length, then runs the same chain of computations. Entering an area works the same way: s = sqrt(4A / (8 * (1 + sqrt(2)))) which simplifies to s = sqrt(A / (2 * (1 + sqrt(2))).

According to Wolfram MathWorld, the area of a regular octagon is 2(1 + sqrt(2)) s^2, where s is the side length.

If the polygon in your problem has six equal sides instead of eight, Hexagon Calculator solves the same set of area, apothem, and perimeter questions with the n = 6 form of the formula.

These four ideas describe what a regular octagon is and how its measurements stay linked. Knowing them helps you read octagon answers with confidence.

These four quantities are tied together by the octagon's central angle. Doubling the side length doubles the apothem, the circumradius, and the perimeter, but it quadruples the area because the area scales with the side length squared.

Because a regular octagon can be split into eight identical isoceles triangles meeting at the center, Triangle Calculator is a useful sanity check when you are solving the central triangle by hand.

Pick the measurement you already have, type it in, and read off the other values. The page updates in real time as you type.

1. 1 Choose a known input: Type a value into Side Length, Apothem, or Area. Side length is the most direct input for the area formula.
2. 2 Watch the chain update: The page converts your input into the side length, then computes the remaining outputs without you pressing Calculate.
3. 3 Cross-check the result: Compare the area, perimeter, and apothem to confirm the unit scale makes sense for the problem at hand.
4. 4 Use the value in your project: Copy the number into a worksheet, CAD file, or assignment. The same square units as your input stay valid throughout.

If a stop sign has a side length of 12 inches, type 12 in the Side Length field. The result panel shows area = 695.2935 sq in, perimeter = 96 in, apothem = 14.4853 in, and circumradius = 15.6788 in, which is enough to estimate the panel area and the radius of the surrounding circle for a sign blank.

When the shape description is loose, Area Calculator lets you start from length and width or radius instead, which is a good fallback for non-regular figures.

The calculator reduces the trig overhead of regular octagons and keeps the linked measurements in sync.

• • One input, four outputs: Side length, apothem, area, or any one of them is enough to derive the rest, so you save time when only one value is given.
• • Fewer rounding mistakes: The cot(pi/8) constant is computed once in full double precision, which avoids cumulative rounding from repeated intermediate trig.
• • Bidirectional support: Apothem and area entries back-solve the side length automatically, so the page works the same way in homework and design workflows.
• • Geometry-friendly units: The square-unit output matches whatever length unit you use, which is convenient for metric and imperial projects alike.
• • Pairs with other polygon tools: The same shape chain works for hexagons, heptagons, and decagons, so you can compare the cost of choosing a different polygon family.

For geometry students the main win is that the closed-form formula and the constant are both shown on the page, so the calculator doubles as a learning aid. For drafters and pattern makers the win is keeping apothem, perimeter, and circumradius in lockstep with the side length while they iterate on a design.

When an octagon is the inner ring of a circular design, Circle Calculator provides the radius, diameter, and circumference of the enclosing circle so the spacing between the two shapes stays consistent.

Four factors change the octagon area you compute, and a few practical limits apply to the regular-octagon assumption.

• • Irregular octagons are not handled. An octagon with sides or angles that differ from the regular form needs a vertex-coordinate method such as the shoelace formula.
• • The calculator uses Euclidean geometry. Curved-surface octagons in spherical or hyperbolic geometry are not in scope.

If you suspect the shape is not regular, measure two or three sides to confirm they match. When the shape is irregular, the polygon area calculator on this site can accept a vertex list and return a more accurate area.

According to Wikipedia, an octagon is an eight-sided polygon, and a regular octagon has interior angles of 135 degrees.

If you need the same kind of area, apothem, and perimeter chain for a ten-sided shape, Decagon Area Calculator runs the n = 10 form of the same regular-polygon formula.