Regular Polygon Area Calculator
Calculate the area of any regular polygon from its side length, apothem, or circumradius. Enter the number of sides and one known dimension to get the area, perimeter, and angles.
Find the Area of a Regular Polygon
Regular Polygon Area
Where:
- \(n\) — number of sides (\(n \ge 3\))
- \(s\) — length of one side
- \(a\) — apothem (distance from the center to the midpoint of a side, i.e. the inradius)
- \(R\) — circumradius (distance from the center to a vertex)
- \(P = n \cdot s\) — perimeter
- \(\pi \approx 3.14159\)
All three expressions give the same area; you use whichever matches the dimension you know. The middle form, \(A = \tfrac{1}{2}Pa\), is the classic result obtained by splitting the polygon into \(n\) identical triangles, each with base \(s\) and height \(a\).
Regular Polygon Area – Calculation Example
Example: a regular hexagon with a side length of 10 cm.
\[ a = \frac{s}{2\tan(\pi/n)} = \frac{10}{2\tan(\pi/6)} = \frac{10}{2 \cdot 0.5774} \approx 8.660 \text{ cm} \]
\[ P = n \cdot s = 6 \cdot 10 = 60 \text{ cm} \]
\[ A = \tfrac{1}{2}\,P\,a = \tfrac{1}{2} \cdot 60 \cdot 8.660 \approx 259.81 \text{ cm}^2 \]
Result: the area is approximately 259.81 cm². The calculator also reports the circumradius \(R = 10\) cm and an interior angle of 120°.
How to Use the Calculator
- Enter the Number of sides (n) — any whole number from 3 (triangle) up to 10,000.
- Choose the dimension you know in the Known quantity dropdown: Side length (s), Apothem (a), or Circumradius (R). Only the matching input field stays visible.
- Type the value in the field that appears — Side length (s), Apothem (a), or Circumradius (R) — and pick its unit (m, cm, mm, ft, or in).
- Read the results: Area (in the square of your unit), plus Perimeter, Side length, Apothem, Circumradius, Interior angle, and a step-by-step derivation.
What Is a Regular Polygon?
A regular polygon is a closed, straight-sided figure in which every side has the same length and every interior angle is equal. Familiar examples are the equilateral triangle (\(n=3\)), the square (\(n=4\)), the regular pentagon (\(n=5\)), and the regular hexagon (\(n=6\)). Because all sides and angles match, a regular polygon is fully determined by just two numbers: the number of sides and one linear dimension.
The area formula works by slicing the polygon into \(n\) congruent triangles that meet at the center. Each triangle has a base \(s\) (one side of the polygon) and a height \(a\) (the apothem), so its area is \(\tfrac{1}{2}sa\). Multiplying by \(n\) sides gives \(A = \tfrac{1}{2}na s = \tfrac{1}{2}Pa\). This is the same logic used to find the area of a circle in the limit: as \(n\) grows very large, the polygon becomes indistinguishable from a circle of radius \(R\), and the formula approaches \(\pi R^2\).
Two derived dimensions are worth understanding. The apothem is the radius of the inscribed circle — the largest circle that fits inside the polygon, touching every side at its midpoint. The circumradius is the radius of the circumscribed circle that passes through every vertex. The three dimensions are linked by the identities \(s = 2a\tan(\pi/n) = 2R\sin(\pi/n)\) and \(a = R\cos(\pi/n)\), which is why the calculator can derive all of them from any single one.
The interior angle shown in the results comes from \(\alpha = (n-2) \cdot 180°/n\). It is useful in construction, tiling, and woodworking: for example, a hexagon has 120° corners, so you cut boards at a 60° miter to join them.
Assumptions, Limitations, and Tips
- Regular polygons only. The formulas rely on all sides and angles being equal. For an irregular polygon, split it into triangles or trapezoids and add the areas.
- Units must be consistent. The unit dropdown converts the input automatically, and the area is returned in the corresponding square unit (e.g. cm → cm²). If you work by hand, keep every length in the same unit before applying the formula.
- Choose the dimension you actually measured. In practical work you often know the distance across flats (twice the apothem, for even \(n\)) or the bolt-circle radius (circumradius) rather than a side — pick that option to avoid rounding errors from an intermediate conversion.
- Large \(n\). Values up to 10,000 are accepted; for such figures the result is essentially the area of a circle with radius \(R\), since \(A \to \pi R^2\) as \(n \to \infty\).
- Rounding. Results are shown to three decimal places. Intermediate values in the calculation steps are rounded for display, so a hand check may differ in the last digit.
Frequently Asked Questions
How do I find the area if I only know the perimeter?
Combine the formulas: since \(s = P/n\), the area is \(A = P^2 / (4n\tan(\pi/n))\). For example, a regular pentagon with perimeter 50 cm gives \(A = 2500 / (20\tan 36°) \approx 172.05\text{ cm}^2\). Enter the side length \(s = P/n\) in the calculator to get the same result along with the apothem and angles.
Why does my hand calculation give a different answer?
The most common mistake is using degrees where radians are required (or vice versa). The formula \(A = ns^2/(4\tan(\pi/n))\) expects the tangent of an angle in radians. If your calculator is in degree mode, use \(\tan(180°/n)\) instead. Also check that the number of sides and the side length are not swapped, and that all lengths are in the same unit before computing.
How do I use the distance across flats, like on a hexagon nut?
The distance across flats is twice the apothem for polygons with an even number of sides. Measure it, divide by 2, and enter that value as the apothem \(a\). For a hexagon with 20 mm across flats, the apothem is 10 mm, and the calculator returns an area of about 259.81 mm². Using the dimension you actually measured avoids rounding errors from converting it to a side length yourself.
Are there simpler formulas for common shapes?
Yes. A regular hexagon with side \(s\) has area \(\frac{3\sqrt{3}}{2}s^2 \approx 2.598\,s^2\). A regular octagon has area \(2(1+\sqrt{2})\,s^2 \approx 4.828\,s^2\). A square is simply \(s^2\) and an equilateral triangle \(\frac{\sqrt{3}}{4}s^2 \approx 0.433\,s^2\). These are all special cases of the general formula, so they are useful for quick sanity checks.
Does a regular polygon have the largest area for a given perimeter?
Among all polygons with the same number of sides and the same perimeter, yes — the regular one encloses the maximum area. This is why the formula gives an upper bound: any irregular polygon with the same \(n\) and perimeter will have a smaller area. In the limit, as the number of sides grows, a regular polygon approaches the circle, which is the most area-efficient shape of all.
Can I use this calculator for irregular polygons or circles?
No — the formulas only hold when every side and angle is equal. For an irregular polygon, split it into triangles or rectangles, compute each area separately, and add them up; a shoelace formula from vertex coordinates also works. For a circle, use \(A = \pi r^2\) directly. A regular polygon with a very large number of sides is nearly a circle, so entering a big \(n\) approximates one.
How can I sanity-check the result?
Compare the polygon's area with two benchmarks: the inscribed circle \(\pi a^2\) is always slightly smaller, and the circumscribed circle \(\pi R^2\) is always slightly larger. Your answer should fall between them and approach both as \(n\) grows. Also verify the interior angle: it must be less than 180° and equal \((n-2)\cdot 180°/n\) — for instance, 108° for a pentagon and 135° for an octagon.