Octagon calculator

What is an octagon calculator?

The octagon calculator is a single tool that describes a regular octagon — the eight-sided shape with equal sides and equal angles, the same outline as a traffic stop sign. Enter one measurement and it returns every other quantity at once: the side length, the area, the perimeter, all three diagonals, the circumradius, and the inradius. It is handy for students checking geometry homework, makers cutting an octagonal frame or tabletop, and anyone laying out a gazebo, a paving pattern, or a sign.

Properties of a regular octagon

A regular octagon has eight equal sides and eight interior angles of 135 degrees each. Because eight vertices are not all the same distance apart, an octagon has three distinct diagonals rather than the two of a hexagon:

• Longest diagonal joins two opposite vertices and passes through the center; it is the full width across the shape.
• Medium diagonal joins two vertices with two vertices between them.
• Shortest diagonal joins two vertices that skip a single vertex.

The circumradius is the distance from the center to any corner, and the inradius (also called the apothem) is the distance from the center to the midpoint of any side.

How does the calculator work?

Type a value into any field and the calculator first recovers the side length from it, then fills in every remaining property. So you can start from the side, the area, the perimeter, any of the three diagonals, the circumradius, or the inradius, and you will always get a complete description of the octagon. Each length field accepts different units, and the conversions between them happen automatically.

Formulas

With side length $a$, the area of a regular octagon is:

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

The perimeter is eight times the side:

$$P = 8a$$

The three diagonals — longest $D$, medium $M$, and shortest $d$ — are:

$$D = a\sqrt{4 + 2\sqrt{2}} \qquad M = a\left(1 + \sqrt{2}\right) \qquad d = a\sqrt{2 + \sqrt{2}}$$

The circumradius $R$ is half the longest diagonal, and the inradius $r$ (the apothem) is half the medium diagonal:

$$R = \frac{a}{2}\sqrt{4 + 2\sqrt{2}} \qquad r = \frac{a\left(1 + \sqrt{2}\right)}{2}$$

where $A$ is the area, $P$ the perimeter, $D$, $M$, and $d$ the longest, medium, and shortest diagonals, $R$ the circumradius, $r$ the inradius, and $a$ the side length.

Examples

1. A regular octagon with a side of 5 cm:

$$A = 2\left(1 + \sqrt{2}\right)\times 5^2 \approx 120.71 \text{ square centimeters}$$ $$P = 8 \times 5 = 40 \text{ centimeters}$$ $$D = 5\sqrt{4 + 2\sqrt{2}} \approx 13.07 \text{ centimeters} \qquad M = 5\left(1 + \sqrt{2}\right) \approx 12.07 \text{ centimeters}$$ $$d = 5\sqrt{2 + \sqrt{2}} \approx 9.24 \text{ centimeters}$$ $$R \approx 6.53 \text{ centimeters} \qquad r \approx 6.04 \text{ centimeters}$$

2. Working backwards from a perimeter of 40 cm, the side is $40 / 8 = 5$ cm, which reproduces all of the values above.

Practical notes

• The longest diagonal is the full span across a flat-sided octagon, so it is the diameter of the smallest circle that contains the shape; the circumradius is exactly half of it.
• The inradius is the apothem — the radius of the largest circle that fits inside the octagon — and is useful when fitting an octagon around a round object.
• For shapes with a different number of sides, the regular polygon area calculator generalizes the area formula, and the hexagon calculator handles the six-sided case.

FAQs

How do I find the area of a regular octagon?

Square the side length and multiply by $2\left(1 + \sqrt{2}\right)\approx 4.8284$. For a side of 5 the area is $2\left(1 + \sqrt{2}\right)\times 25 \approx 120.71$.

What is the difference between the three diagonals?

The longest diagonal joins opposite vertices and runs through the center, equal to $a\sqrt{4 + 2\sqrt{2}}$. The medium diagonal skips two vertices and equals $a\left(1 + \sqrt{2}\right)$. The shortest diagonal skips one vertex and equals $a\sqrt{2 + \sqrt{2}}$.

What is the apothem of an octagon?

The apothem is the inradius — the distance from the center to the middle of a side. For a regular octagon it equals $\frac{a\left(1 + \sqrt{2}\right)}{2}$, about 1.207 times the side.

How wide is a regular octagon?

The width across opposite sides is twice the inradius, $a\left(1 + \sqrt{2}\right)$, which is also the medium diagonal. The width across opposite corners is the longest diagonal, $a\sqrt{4 + 2\sqrt{2}}$.