Regular Polygon Calculator

Introduction

A regular polygon is a shape with all sides equal and all interior angles equal: the equilateral triangle, the square, the pentagon, the hexagon, the octagon, and so on. That single symmetry condition is remarkably restrictive. Once you fix the number of sides and the length of one side, every other property of the shape is determined — the perimeter, the area, the apothem (the distance from the centre to a side), the circumradius (the distance from the centre to a vertex), and both the interior and exterior angles [mathworld-regular-polygon]. There is nothing left to choose.

That determinism is what makes regular polygons so useful. A hexagonal tile, an octagonal tabletop, a pentagonal paving stone and a square napkin are all specified by two numbers, and everything a builder, designer or student needs follows from those two. The difficulty is that the relationships between those properties are trigonometric rather than arithmetic: the apothem involves a tangent, the circumradius a sine, and the area combines both with the perimeter. This calculator does that trigonometry for you and returns every property at once.

The most common practical need is area. Ordering material for a hexagonal patio means knowing how many square feet of paving you need; cutting an octagonal tabletop means knowing how much stock to buy; coating a pentagonal sign means knowing the surface to paint. Perimeter is the companion figure: it drives edging, framing, fencing and the length of trim required. The apothem matters whenever the shape must be inscribed in or circumscribed around something else — a hexagonal nut inside a circular bore, or a square inscribed in a circle of known diameter [khan-geo].

The angles are the other half of the story. The interior angle of a regular polygon grows with the number of sides, approaching 180° as the polygon approaches a circle: 60° for a triangle, 90° for a square, 108° for a pentagon, 120° for a hexagon, 135° for an octagon. Anyone tiling a surface or joining pieces at an angle needs that number, because the joint between adjacent pieces is determined by it. The exterior angle — 360° divided by the number of sides — is what actually controls how much you rotate between pieces, and it is the number used when laying out a shape around a point.

This calculator works for any regular polygon with three or more sides, from the triangle up to figures with hundreds of sides where the shape is visually indistinguishable from a circle. It reports the area, perimeter, apothem, circumradius, interior angle, exterior angle and central angle, along with the working so the result can be checked. For irregular shapes, the Area Calculator and the Triangle Calculator cover the general cases.

How to Use

Enter the number of sides and the length of one side. Both are integers or decimals in any unit — the calculator works in whatever unit you enter and does not convert.

The Formula

Every property of a regular polygon with $n$ sides of length $s$ follows from the central angle of one of its triangular sectors, $\theta = \frac{360^\circ}{n}$. Splitting the half-sector gives a right triangle whose opposite side is $s/2$ and whose angle at the centre is $\theta/2$, which yields the apothem and the circumradius:

The perimeter is simply the side length repeated:

And the area follows from the fact that the polygon can be cut into $n$ identical isosceles triangles, each with base $s$ and height $a$:

The angles come from the fact that the interior angles of any $n$-gon sum to $(n-2) \times 180^\circ$, and a regular polygon divides that sum equally:

For a hexagon with side 5: $\theta = 60^\circ$, $a = \frac{5}{2\tan 30^\circ} \approx 4.3301$, $P = 30$ and $A = \frac{1}{2} \times 30 \times 4.3301 \approx 64.9519$.

Reference Table

The table lists the regular polygons that appear most often in practice, with the interior angle and the area for a unit side length:

Two patterns are worth noting. The interior angle rises towards 180° and the exterior angle falls towards 0° as the number of sides increases, and the two always sum to 180° — they are supplementary at every vertex, which is a useful check on the arithmetic. The area, by contrast, grows without limit: a polygon with a fixed side length and more sides encloses more space, approaching the area of the circumscribed circle. A 20-sided figure with unit sides has more than thirty times the area of a square with unit sides [wolfram-geo].

Practical Tips

Use the square as a validation case. Enter 4 sides and any length. The area must come out exactly equal to the side squared and the apothem exactly half the side. This single test catches almost every input error before it reaches a real project.

Remember that the apothem is the inradius. The distance from the centre to a side is the radius of the largest circle that fits inside the polygon, and the circumradius is the radius of the smallest circle that contains it. When a hexagonal part must fit inside a round bore, the apothem is the number that matters, not the circumradius [nctm].

Check the angle sum before trusting a shape. The interior angles of any polygon with $n$ sides sum to $(n-2) \times 180^\circ$: 540° for a pentagon, 720° for a hexagon, 1080° for an octagon. Multiplying the reported interior angle by the number of sides must reproduce that sum.

Use the exterior angle for layout. When stepping around a shape with a compass, protractor or a rotating jig, the rotation between consecutive sides is the exterior angle, $360^\circ/n$. For a hexagon that is a clean 60°; for a heptagon it is the awkward 51.43°.

Only three regular polygons tile the plane. Triangles, squares and hexagons tile without gaps because their interior angles divide 360° evenly: $60^\circ \times 6 = 360^\circ$, $90^\circ \times 4 = 360^\circ$ and $120^\circ \times 3 = 360^\circ$. No other regular polygon does, which is why pentagons and octagons always need filler pieces in a tiling.

Scale by the square of the side. The area scales with $s^2$, so doubling the side length quadruples the area and doubling the material order. Perimeter, by contrast, only doubles — a useful reminder when a design is scaled up.

Watch the tangent near small angles. For a polygon with many sides, $180^\circ/n$ becomes very small and the tangent very close to the angle in radians, so the apothem approaches the circumradius. That is the mathematical reason a many-sided polygon looks like a circle [nist-geo].

Limitations

This calculator handles regular polygons only — shapes where every side and every angle is equal. For an irregular polygon, split it into triangles or use the Area Calculator, which accepts arbitrary shapes.

Inputs must be a side count of at least 3 and a positive side length. A side count below 3 is not a polygon, and a non-positive side length has no geometric meaning. Decimal side lengths are accepted, and the results inherit the unit you enter.

The results are computed with floating-point arithmetic and displayed to four decimal places. For very large side counts the apothem, circumradius and area converge, and rounding may make the printed apothem appear identical to the circumradius even though they differ mathematically.

Angles are reported in degrees only. If radians are needed, multiply by $\pi/180$; the Trigonometry Calculator handles angle conversions and the trigonometric functions directly.

Finally, the calculator reports geometric properties, not material properties. It does not account for cut widths, joint gaps, waste, or the thickness of a coating — all of which must be added on top of the geometric result before ordering material.

Frequently Asked Questions

❓ How do you find the area of a regular polygon?

✅ Multiply the perimeter by the apothem and halve the result. For a hexagon with side 5 the perimeter is 30, the apothem is about 4.3301, and the area is 0.5 x 30 x 4.3301, about 64.95 square units.

❓ What is the apothem?

✅ The perpendicular distance from the centre of the polygon to the midpoint of a side. It is also the radius of the inscribed circle, and it is the height of each of the isosceles triangles that the polygon splits into.

❓ What is the interior angle of a regular hexagon?

✅ 120 degrees. The interior angles of any n-gon sum to (n - 2) x 180 degrees, which for a hexagon is 720 degrees, and a regular hexagon divides that evenly: 720 / 6 = 120.

❓ Which regular polygons tile a plane?

✅ Only the triangle, the square and the hexagon. Their interior angles are 60, 90 and 120 degrees, and each divides 360 degrees exactly, so copies meet edge to edge without gaps. All other regular polygons leave gaps.

❓ How is the apothem calculated?

✅ It is the side length divided by twice the tangent of half the central angle: a = s / (2 tan(180/n degrees)). For a square that is s / (2 tan 45) = s/2, which matches the obvious answer.

❓ What is the difference between the apothem and the circumradius?

✅ The apothem reaches a side, the circumradius reaches a vertex. The circumradius is always the larger of the two, and for a square with side 10 the apothem is 5 while the circumradius is about 7.071.

❓ How do I find the perimeter of a regular polygon?

✅ Multiply the number of sides by the side length. An octagon with side 2.5 has a perimeter of 8 x 2.5 = 20, and the perimeter is the same for any polygon with the same number of sides and side length.

❓ What is the exterior angle and why does it matter?

✅ It is 360 degrees divided by the number of sides, and it is the rotation needed between consecutive sides when laying out the shape. For a hexagon it is 60 degrees, which is why a hexagon can be stepped out with a simple 60-degree rotation.

❓ Does the area formula work for a triangle?

✅ Yes. With 3 sides the formula gives 0.5 x 3s x s/(2 tan 60), which reduces to the familiar equilateral triangle area (sqrt(3)/4) s squared.

❓ How does the area change if I double the side length?

✅ It quadruples. Area scales with the square of the side length, so doubling the side multiplies the area by four, while the perimeter only doubles.

❓ Can I use this for an irregular polygon?

✅ No. The formulas rely on every side and every angle being equal. For irregular shapes, divide the figure into triangles, or use the Area Calculator, which accepts arbitrary shapes.

❓ Why does a many-sided polygon look like a circle?

✅ As the number of sides grows, the central angle shrinks, the apothem approaches the circumradius, and the area approaches that of the circumscribed circle. A 20-sided polygon is already visually indistinguishable from a circle at small sizes.