Octagon Calculator

Q: How many sides does an octagon have?

An octagon has 8 sides . In a regular octagon, all 8 sides are the same length and all 8 angles are equal.

Q: What is the formula for the area of a regular octagon?

The area formula is A = 2(1 + √2) × a² , where a is the side length. This is about 4.828 × a² . For example, if the side length is 10, the area is about 482.843 square units.

Q: What is the difference between the short, medium, and long diagonals?

The short diagonal (f) connects two vertices with one vertex between them. The medium diagonal (e) connects two vertices with two vertices between them. The long diagonal (d) goes all the way through the center and connects opposite vertices. The long diagonal is always the longest of the three.

Q: What is the difference between the circumradius and the inradius?

The circumradius (R) is the distance from the center to any corner (vertex) of the octagon. The inradius (r) , also called the apothem, is the distance from the center to the middle of any side. The circumradius is always a bit larger than the inradius.

Q: Does this calculator work for irregular octagons?

No. This calculator only works for regular octagons , where all 8 sides are equal and all 8 angles are equal. If your octagon has sides of different lengths, these formulas will not give correct results.

Q: How do I find the side length if I know the perimeter?

Divide the perimeter by 8. The formula is a = P ÷ 8 . For example, if the perimeter is 80, the side length is 80 ÷ 8 = 10. You can also just type the perimeter into the calculator and it will find the side length for you.

Q: How many diagonals does an octagon have?

A regular octagon has 20 diagonals . You can find this using the formula n(n − 3) ÷ 2 , where n is the number of sides. So it's 8 × (8 − 3) ÷ 2 = 8 × 5 ÷ 2 = 20.

Introduction

An octagon is a shape with eight sides and eight angles. You see octagons every day — the most common example is a stop sign. This octagon calculator helps you find important measurements like side length, area, perimeter, and diagonals. Just enter one known value, and the calculator does the rest of the math for you. It works for regular octagons, which means all eight sides are the same length and all eight angles are equal. Each interior angle of a regular octagon measures 135 degrees. Whether you are working on a homework problem, planning a building project, or just curious about this eight-sided shape, this tool gives you fast and accurate results.

How to Use Our Octagon Calculator

Enter the side length of your regular octagon, and this calculator will instantly find the area, perimeter, diagonal length, and other important measurements for you.

Side Length: Type in the length of one side of your octagon. Since a regular octagon has eight sides that are all the same length, you only need to enter one side. Use any unit you like, such as inches, centimeters, meters, or feet. The results will be in the same unit you choose.

What Is a Regular Octagon?

A regular octagon is a flat shape with 8 equal sides and 8 equal angles. You see octagons in everyday life — the most common example is a stop sign. Because all the sides and angles are the same, a regular octagon has a lot of symmetry, which makes it easy to calculate its properties once you know just one measurement. If you're interested in a similar polygon with fewer sides, try our hexagon calculator for six-sided shapes.

Key Properties of a Regular Octagon

Every regular octagon shares certain fixed angle measurements. Each interior angle is always 135°, and each exterior angle is always 45°. The sum of all interior angles in an octagon is 1,080°. A regular octagon also has exactly 20 diagonals — lines that connect non-adjacent vertices inside the shape. You can verify the number of diagonals using the combination calculator, since diagonal count is based on choosing 2 vertices from 8 and subtracting the 8 sides.

Understanding the Different Measurements

A regular octagon has several important measurements, all of which can be figured out from the side length (a) alone:

Perimeter (P): The total distance around the octagon. Since there are 8 equal sides, the perimeter is simply P = 8a.
Area (A): The space inside the octagon. The formula is A = 2(1 + √2) × a², which is roughly 4.828 × a². For simpler rectangular spaces, you might use the area of a rectangle calculator or the square footage calculator.
Short Diagonal (f): A line connecting two vertices that have one vertex between them. Its length is a × √(2 + √2), or about 1.848 × a.
Medium Diagonal (e): A line connecting two vertices with two vertices between them. It equals a × (1 + √2), or about 2.414 × a. This is also the height of the octagon measured across a flat side.
Long Diagonal (d): The longest diagonal, which passes straight through the center and connects opposite vertices. Its length is a × √(4 + 2√2), or about 2.613 × a. You can use our distance calculator to verify diagonal lengths between vertex coordinates.
Circumradius (R): The radius of the circle that passes through all 8 vertices (the circumscribed circle). It equals half the long diagonal. For calculations involving the circumscribed circle itself, see our circle area calculator.
Inradius or Apothem (r): The radius of the circle that fits perfectly inside the octagon, touching the midpoint of each side (the inscribed circle). Its formula is r = a(1 + √2) / 2, or about 1.207 × a.

How This Calculator Works

This octagon calculator lets you enter any single known property — side length, perimeter, area, any diagonal, circumradius, or inradius — and it instantly computes all the other properties for you. Because every measurement in a regular octagon is directly tied to the side length through a fixed ratio, knowing just one value is enough to determine everything else. Simply type a number into any field, and the calculator does the rest. The area formula involves a square root, and if you need help with those kinds of expressions you can check out our exponent calculator or sig fig calculator to manage precision in your results.

Where Regular Octagons Appear

Beyond stop signs, regular octagons show up in architecture, tile patterns, flooring designs, umbrella frames, and fortress layouts. Architects and engineers use octagonal shapes because they offer a good balance between a circle and a square — they enclose a large area relative to their perimeter while still having flat sides that are easy to build. If you're working on a construction project involving octagonal tiles or pavers, our tile calculator and paver calculator can help you estimate materials. For related geometry problems involving triangles formed by diagonals inside an octagon, the right triangle calculator, triangle area calculator, and law of cosines calculator are useful companions. Understanding octagon geometry is valuable for planning construction projects, cutting materials, or solving math problems in school.

Frequently Asked Questions

How many sides does an octagon have?

An octagon has 8 sides. In a regular octagon, all 8 sides are the same length and all 8 angles are equal.

What is the formula for the area of a regular octagon?

The area formula is A = 2(1 + √2) × a², where a is the side length. This is about 4.828 × a². For example, if the side length is 10, the area is about 482.843 square units.

Can I use this calculator if I only know the area?

Yes. Enter the area into the Area field, and the calculator will figure out the side length and all other properties automatically. You can start from any single known value — side, perimeter, area, any diagonal, circumradius, or inradius.

What is the difference between the short, medium, and long diagonals?

The short diagonal (f) connects two vertices with one vertex between them. The medium diagonal (e) connects two vertices with two vertices between them. The long diagonal (d) goes all the way through the center and connects opposite vertices. The long diagonal is always the longest of the three.

What is the difference between the circumradius and the inradius?

The circumradius (R) is the distance from the center to any corner (vertex) of the octagon. The inradius (r), also called the apothem, is the distance from the center to the middle of any side. The circumradius is always a bit larger than the inradius.

Does this calculator work for irregular octagons?

No. This calculator only works for regular octagons, where all 8 sides are equal and all 8 angles are equal. If your octagon has sides of different lengths, these formulas will not give correct results.

What units does the octagon calculator use?

The calculator does not use a fixed unit. You can enter your value in any unit — inches, centimeters, meters, feet, or anything else. The results will be in the same unit you used. For area, the result will be in that unit squared (e.g., square inches or square meters).

How do I find the side length if I know the perimeter?

Divide the perimeter by 8. The formula is a = P ÷ 8. For example, if the perimeter is 80, the side length is 80 ÷ 8 = 10. You can also just type the perimeter into the calculator and it will find the side length for you.

Why does the interior angle always equal 135 degrees?

The interior angle of any regular polygon depends only on the number of sides. The formula is (n − 2) × 180° ÷ n. For an octagon, n = 8, so it's (8 − 2) × 180° ÷ 8 = 135°. This is the same for every regular octagon, no matter how big or small.

How many diagonals does an octagon have?

A regular octagon has 20 diagonals. You can find this using the formula n(n − 3) ÷ 2, where n is the number of sides. So it's 8 × (8 − 3) ÷ 2 = 8 × 5 ÷ 2 = 20.

Is the long diagonal the same as the circumradius?

No, but they are related. The long diagonal (d) passes all the way through the center, connecting two opposite vertices. The circumradius (R) is exactly half the long diagonal, since it goes from the center to just one vertex. So d = 2R.

How accurate are the results from this calculator?

The results are very accurate. The calculator uses precise mathematical constants and formulas, and displays results rounded to up to 6 decimal places. This level of precision is more than enough for homework, design work, and most construction projects.

What is an apothem?

The apothem is the distance from the center of a regular polygon to the midpoint of one of its sides. In a regular octagon, it is the same as the inradius (r). It equals a × (1 + √2) ÷ 2, or about 1.207 times the side length.

How do I find the area of an octagon from the long diagonal?

First, find the side length from the long diagonal using a = d ÷ √(4 + 2√2). Then use the area formula A = 2(1 + √2) × a². Or simply type the long diagonal into the calculator and it does both steps for you instantly.