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Regular Polygon Calculator
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What Are Regular Polygon Measurements?
A regular polygon is a flat, closed shape with three or more straight sides where every side has the same length and every interior angle has the same measure. An equilateral triangle, square, regular pentagon, and regular hexagon are all examples of regular polygons.
Regular polygon measurements are connected. Once you know the number of sides and one suitable measurement, such as the side length, apothem, circumradius, perimeter, or area, the other common measurements can be found from geometry and trigonometry.
This matters because a regular polygon has a predictable structure. Instead of measuring every side or angle separately, you can use the symmetry of the shape. The center, vertices, and side midpoints form repeated right triangles, and those right triangles make the formulas work.
For most school and everyday geometry uses, the formulas here apply to the usual simple, convex regular polygons: shapes that do not cross themselves and have all vertices pointing outward.
Why Regular Polygon Measurements Matter
Regular polygons appear in geometry classes, tiling patterns, design layouts, architecture, games, logos, machining, and construction plans. Even when the final object is not a perfect regular polygon, regular polygon formulas are often used for sketches, estimates, and symmetric designs.
They are also useful because many circular ideas can be approximated with regular polygons. As the number of sides increases, a regular polygon begins to look more like a circle. This is one reason apothem, circumradius, perimeter, and area are often studied together.
Understanding the relationships between these measurements helps you avoid common mistakes, such as using the radius to a vertex when the formula requires the apothem, or entering an area value while treating it as a length.
Key Terms to Know
- Regular polygon: A polygon with all sides equal and all interior angles equal.
- Number of sides, \(n\): The whole-number count of sides in the polygon. A polygon must have at least \(3\) sides.
- Side length, \(s\): The length of one side. In a regular polygon, every side has this same length.
- Perimeter, \(P\): The total distance around the polygon.
- Area, \(A\): The amount of flat surface enclosed by the polygon, measured in square units.
- Apothem, \(a\): The distance from the center of the polygon to the midpoint of a side. It meets the side at a right angle.
- Circumradius, \(R\): The distance from the center of the polygon to a vertex.
- Central angle: The angle at the center between two neighboring vertices.
- Interior angle: The angle inside the polygon at each vertex.
How Regular Polygon Measurements Work
A regular polygon can be divided into \(n\) matching isosceles triangles by drawing segments from the center to each vertex. Each of those triangles has a central angle of:
$$
\frac{360^\circ}{n}
$$
If you split one of those isosceles triangles down the middle, you get a right triangle. In that right triangle:
- the half-side is \(\frac{s}{2}\),
- the apothem \(a\) is one leg,
- the circumradius \(R\) is the hypotenuse,
- the center angle is \(\frac{\pi}{n}\) radians.
That right triangle gives the main trigonometric relationships:
$$
s = 2R\sin\left(\frac{\pi}{n}\right)
$$
$$
s = 2a\tan\left(\frac{\pi}{n}\right)
$$
Solving those for \(R\) and \(a\) gives:
$$
R = \frac{s}{2\sin\left(\frac{\pi}{n}\right)}
$$
$$
a = \frac{s}{2\tan\left(\frac{\pi}{n}\right)}
$$
The perimeter is the number of sides times the side length:
$$
P = ns
$$
The area can be found by treating the polygon as \(n\) triangles, each with base \(s\) and height \(a\). This gives:
$$
A = \frac{1}{2}Pa
$$
Since \(P = ns\), the same area formula can also be written as:
$$
A = \frac{1}{2}nsa
$$
If the area is the known value and the side length is unknown, the side length can be found from:
$$
s = \sqrt{\frac{4A\tan\left(\frac{\pi}{n}\right)}{n}}
$$
The angle formulas are separate from the size of the polygon. They depend only on the number of sides:
$$
\text{central angle} = \frac{360^\circ}{n}
$$
$$
\text{interior angle} = \frac{(n - 2)180^\circ}{n}
$$
These formulas explain why one known measurement is enough for a regular polygon. Once the side count and one size measurement are known, the side length can be derived, and the rest of the values follow from it.
Examples of Regular Polygon Measurements in Practice
Example 1: Regular Hexagon From Side Length
Suppose a regular hexagon has \(6\) sides and each side is \(10\text{ cm}\).
First find the perimeter:
$$
P = ns = 6 \times 10 = 60\text{ cm}
$$
Find the apothem:
$$
a = \frac{10}{2\tan\left(\frac{\pi}{6}\right)} \approx 8.6603\text{ cm}
$$
Find the circumradius:
$$
R = \frac{10}{2\sin\left(\frac{\pi}{6}\right)} = 10\text{ cm}
$$
Find the area:
$$
A = \frac{1}{2}Pa = \frac{1}{2}(60)(8.6603) \approx 259.81\text{ cm}^2
$$
The central angle is:
$$
\frac{360^\circ}{6} = 60^\circ
$$
The interior angle is:
$$
\frac{(6 - 2)180^\circ}{6} = 120^\circ
$$
A regular hexagon is a useful special case because its side length equals its circumradius.
Example 2: Regular Octagon From Circumradius
Suppose a regular octagon has a circumradius of \(5\text{ in}\). The circumradius is the distance from the center to a vertex, not the full distance across the polygon.
Find the side length:
$$
s = 2R\sin\left(\frac{\pi}{n}\right)
$$
$$
s = 2(5)\sin\left(\frac{\pi}{8}\right) \approx 3.8268\text{ in}
$$
Find the perimeter:
$$
P = 8(3.8268) \approx 30.6147\text{ in}
$$
Find the apothem:
$$
a = R\cos\left(\frac{\pi}{8}\right) \approx 4.6194\text{ in}
$$
Find the area:
$$
A = \frac{1}{2}(30.6147)(4.6194) \approx 70.7107\text{ in}^2
$$
The central angle is \(45^\circ\), and each interior angle is \(135^\circ\).
Example 3: Regular Pentagon From Area
Suppose a regular pentagon has an area of \(100\text{ cm}^2\). Because the known value is an area, the unit is square centimeters, not centimeters.
Use the area-to-side formula:
$$
s = \sqrt{\frac{4A\tan\left(\frac{\pi}{n}\right)}{n}}
$$
Substitute \(A = 100\) and \(n = 5\):
$$
s = \sqrt{\frac{4(100)\tan\left(\frac{\pi}{5}\right)}{5}} \approx 7.6239\text{ cm}
$$
Then the perimeter is:
$$
P = 5(7.6239) \approx 38.1194\text{ cm}
$$
The apothem is:
$$
a = \frac{7.6239}{2\tan\left(\frac{\pi}{5}\right)} \approx 5.2467\text{ cm}
$$
This example shows why the known value type matters. Entering \(100\) as a side length would describe a much larger pentagon than entering \(100\text{ cm}^2\) as an area.
How to Interpret the Result
A regular polygon result usually contains several different kinds of measurements. They are related, but they do not mean the same thing.
- Side length is the length of one side only.
- Perimeter is the total distance around all sides.
- Area is the space inside the polygon and is always measured in square units.
- Apothem is measured from the center to the midpoint of a side.
- Circumradius is measured from the center to a vertex.
- Interior angle is the angle at each corner inside the polygon.
- Central angle is the angle at the center between adjacent vertices.
A larger side length, apothem, circumradius, perimeter, or area means the polygon is larger in scale. A larger number of sides changes the shape as well as the formulas: for the same circumradius, more sides make the polygon closer to a circle.
Angle results behave differently from length and area results. The interior angle gets larger as the number of sides increases, while the central angle gets smaller. A triangle has a central angle of \(120^\circ\), a square has \(90^\circ\), and a regular hexagon has \(60^\circ\).
Numerical results are best treated as rounded decimal values unless the values are simple enough to be exact. Rounding is especially noticeable when trigonometric functions produce long decimals.
Common Mistakes and Misconceptions
Confusing apothem and circumradius. The apothem goes from the center to the middle of a side. The circumradius goes from the center to a vertex. The circumradius is usually longer than the apothem for the same regular polygon.
Entering a diameter as a circumradius. A diameter is the full distance across a circle or across opposite points through the center. A circumradius is only half that distance. If you have a diameter, divide it by \(2\) before using it as a circumradius.
Using area mode with a length unit. Area must be entered in square units, such as \(\text{cm}^2\) or \(\text{ft}^2\). A value of \(25\text{ cm}\) and a value of \(25\text{ cm}^2\) describe different kinds of measurements.
Expecting fractional side counts to work. A polygon side count is a whole number. Values such as \(5.5\) sides do not describe a standard polygon.
Using regular polygon formulas for irregular polygons. If the sides or angles are not all equal, these formulas do not apply. Irregular polygons need different information, such as all side lengths, coordinates, or a decomposition into simpler shapes.
Mixing units without converting. If one measurement is in inches and another is in centimeters, convert first. A calculation should use one consistent unit system at a time.
Rounding too early. For best accuracy, keep extra digits during intermediate steps and round only the final result.
When to Use Regular Polygon Formulas
Use regular polygon formulas when:
- the shape has equal side lengths and equal interior angles;
- you know the number of sides and one supported measurement;
- you need side length, perimeter, area, apothem, circumradius, or angle measures;
- you are checking geometry homework or teaching polygon relationships;
- you are making a quick estimate for a symmetric design or layout;
- you want to understand how a many-sided polygon compares with a circle.
Do not use these formulas when the polygon is irregular, self-crossing, measured from uneven real-world points, or defined by information such as diagonals or coordinates rather than a regular side count and one standard measurement.
Limitations and Things to Keep in Mind
These formulas assume a regular polygon. That means all sides are equal and all interior angles are equal. If the shape is only approximately regular, the result is only an approximation.
The number of sides must be a whole number greater than or equal to \(3\). A known length, perimeter, radius, apothem, or area must be positive. Zero and negative values do not describe a physical regular polygon size.
Supported known value types are limited to side length, circumradius, apothem, perimeter, and area. Other possible measurements, such as diameter, diagonal length, exterior angle, or a custom angle, must be converted or handled separately.
Supported length units are millimeters, centimeters, meters, kilometers, inches, and feet. Area values use the squared versions of those units. Units such as yards, miles, and mixed feet-and-inches inputs are not supported directly.
Displayed numeric values are rounded to up to \(8\) significant digits, with unnecessary trailing zeros removed. Very tiny values may display as \(0\). Results that cannot be represented in the selected unit are rejected with a range warning rather than shown as partial results.
The diagram is a visual aid, not a scaled technical drawing. For very large side counts, the visual display may be simplified, while the numeric results still use the entered number of sides.
For classroom work, design sketches, or general estimation, these formulas are usually enough. For engineering, construction, manufacturing, safety-critical work, or official documentation, double-check measurements and use the standards or professional tools required for that context.
How to Use This Calculator
- Enter the number of sides, such as \(3\) for a triangle, \(4\) for a square, or \(6\) for a hexagon.
- Choose the known value type: side length, circumradius, apothem, perimeter, or area.
- Enter a positive known value.
- Choose the matching unit. Area mode uses square-unit options.
- Review the calculated side length, perimeter, area, apothem, circumradius, interior angle, and central angle.
- Use the diagram as a visual check, and download the graph as a PNG when you need an image of the polygon.
Frequently Asked Questions
Is the apothem the same as the radius?
Not exactly. The apothem is the distance from the center to the midpoint of a side. The circumradius is the distance from the center to a vertex. In a regular polygon, both start at the center, but they end at different places.
Why does the number of sides have to be at least 3?
A polygon is a closed shape made from straight sides. One or two sides cannot enclose a flat region, so the smallest polygon has \(3\) sides: a triangle.
Can I enter the diameter as the circumradius?
No. Circumradius means the distance from the center to a vertex. If you know the diameter across opposite vertices, divide the diameter by \(2\) before using it as the circumradius.
Why does area use square units?
Area measures two-dimensional space. If length is measured in centimeters, area is measured in square centimeters. This is why \(10\text{ cm}\) is a length, while \(10\text{ cm}^2\) is an area.
Can these formulas be used for an irregular polygon?
No. Irregular polygons do not have one shared side length and one shared interior angle. Their areas and perimeters require different information, such as all side lengths, coordinates, or a breakdown into triangles or other simpler shapes.
What happens when the number of sides is very large?
As the side count increases, a regular polygon looks more like a circle. The central angle becomes smaller, and neighboring vertices become closer together. Numeric formulas can still describe the polygon, but a diagram may not show every side clearly when the side count is very high.
Sources and References
Books
- Henry Africk. Elementary College Geometry (2021 ed.). New York City College of Technology, CUNY Academic Works, 2021. Relevant sections: Chapter 7, “Regular Polygons and Circles,” especially Section 7.1, “Regular Polygons.” https://academicworks.cuny.edu/ny_oers/44/
- Donna Kirk. Contemporary Mathematics. OpenStax, 2023. Relevant sections: Chapter 10, Section 10.4, “Polygons, Perimeter, and Circumference,” and Section 10.6, “Area.” https://openstax.org/books/contemporary-mathematics/pages/1-introduction
Online and Official Sources
- Eric W. Weisstein. “Regular Polygon.” Wolfram MathWorld, accessed July 4, 2026. https://mathworld.wolfram.com/RegularPolygon.html
- National Institute of Standards and Technology. “Guide to the SI, Appendix B: Conversion Factors.” NIST, last updated February 1, 2016. https://www.nist.gov/pml/special-publication-811/nist-guide-si-appendix-b-conversion-factors
- National Institute of Standards and Technology. “SI Units – Length.” NIST, accessed July 4, 2026. https://www.nist.gov/pml/owm/si-units-length