Regular Polygon Calculator

JJ Ben-Joseph headshot Editorial review by: JJ Ben-Joseph

Understand the geometry before you calculate

A regular polygon is a flat shape with a simple but powerful rule: every side has the same length, and every interior angle has the same measure. That one condition lets you compute a surprising amount from just two inputs. With the calculator on this page, you enter the number of sides n and the side length s, and the tool returns the perimeter, area, interior angle, exterior angle, apothem, central angle, and circumradius. In other words, it turns a basic description of a regular polygon into a full geometric profile.

The two inputs do different jobs. The side count tells the calculator which family of shape you mean: 3 for an equilateral triangle, 4 for a square, 5 for a regular pentagon, 6 for a regular hexagon, and so on. The side length sets the scale. If you measure the side in centimeters, then the perimeter, apothem, and circumradius come out in centimeters, while the area comes out in square centimeters. If you measure the side in meters, the same structure holds in meters and square meters. That makes unit handling easy: keep the side length in one consistent unit, and the results stay internally consistent.

Seen from a broad mathematical viewpoint, the calculator still follows the same pattern as many technical tools. It maps a small set of inputs to a result using a rule or formula:

R = f (x_{1}, x_{2}, \dots, x_{n})

And when a calculation combines several related pieces, it can be described as a sum of contributions:

T = \sum_{i = 1}^{n} w_{i} \cdot x_{i}

For this specific tool, those abstract symbols narrow down to familiar polygon quantities. The perimeter is the easiest result because it is just the side length repeated around the shape. If each side has length $s$ and there are $n$ sides, then the perimeter is $P = n s$ . The exterior angle is the turn you make at each corner while tracing the polygon, so all exterior angles add to 360 degrees. That gives the clean formula $A_{e} = \frac{360}{n}$ . Each interior angle is the supplement of that exterior angle, which leads to $A_{i} = \frac{n - 2}{n} \cdot 180 °$ .

The apothem and area come from a useful picture: connect the center of the polygon to every vertex. That splits the figure into n congruent isosceles triangles. The apothem is the distance from the center to the midpoint of a side, so it acts like the height of each triangle. Trigonometry gives the apothem formula

a = \frac{s}{2 \cdot \tan (\frac{π}{n})}

Once the apothem is known, the area follows from the familiar triangle formula. One triangle has area $\frac{1}{2} a s$ , and multiplying by n gives the full polygon area $\frac{1}{2} P a$ . That is why the results panel can move from the side length to a complete area answer without asking for any extra dimensions.

A quick worked example makes the relationships easier to trust. Suppose you enter 6 sides and a side length of 8 units. The perimeter is 6 × 8 = 48 units. The exterior angle is 360 ÷ 6 = 60°, so the interior angle is 120°. The apothem is a little under 6.93 units, and the area is about 166.28 square units. Those numbers tell a consistent story: a hexagon with moderate side length has a perimeter that scales directly with the side count, while the area depends on both the outline and how far the polygon reaches inward toward its center.

When you interpret the result, it helps to do three quick checks. First, confirm that the side count is a whole number of 3 or greater; a regular polygon cannot have 4.5 sides. Second, confirm that the side length is positive and uses the unit you intend. Third, check whether the outputs move in the right direction when you change an input. If you increase the side length, every linear measurement should increase. If you hold the side length fixed and increase the number of sides, the perimeter grows, the exterior angle shrinks, the interior angle grows toward 180°, and the polygon begins to look more and more like a circle.

Regularity matters: the formulas assume all sides and all angles are equal. An irregular polygon needs different methods.
Units carry through the calculation: perimeter, apothem, and circumradius use the same unit as the side length, while area uses square units.
Rounding is normal: the display rounds values, especially for apothem, area, and circumradius when trigonometric values are involved.
Large side counts approach a circle: that is a helpful intuition, but the calculator still uses exact regular polygon formulas rather than circle formulas.

Why regular polygons are worth studying

A regular polygon is one of the most useful meeting points between geometry, algebra, and trigonometry. Because every side and every angle repeat in a predictable way, regular polygons let you explore big ideas with relatively simple arithmetic. In a classroom, they support lessons on angle sum, congruent triangles, symmetry, and limits. Outside the classroom, they appear in engineering drawings, logos, signage, tiling, woodworking, packaging, and computer graphics. The shape is familiar; the mathematics behind it is rich.

The perimeter formula $P = n s$ follows immediately from the definition, since every side has length $s$ . The apothem formula uses trigonometry, because half of one central triangle forms a right triangle with angle $\frac{π}{n}$ . That gives

a = \frac{s}{2 \cdot \tan (\frac{π}{n})}

and from there the area becomes

\frac{1}{2} P a

because the polygon can be treated as a ring of congruent triangles around the center. The angle formulas are just as elegant. Each interior angle is

A_{i} = \frac{n - 2}{n} \cdot 180 °

while each exterior angle is

A_{e} = \frac{360}{n}

and the two are supplementary.

To see those formulas in action, consider a square with side length 5 centimeters. Plugging $n=4$ and $s=5$ into the calculator yields a perimeter of 20 centimeters and each interior angle measuring 90 degrees. The apothem, derived from

a = \frac{5}{2 \cdot \tan (\frac{π}{4})}

simplifies to 2.5 centimeters. Applying the area formula provides

\frac{1}{2} \cdot 20 \cdot 2.5 = 25

square centimeters, which matches the familiar result $5^{2}$ . This is a good reminder that the calculator is not introducing a new kind of geometry; it is packaging the same geometry you may already know into a quick, consistent workflow.

Now compare that square with a 12-sided regular polygon. Setting $n=12$ with side length 4 centimeters produces interior angles of 150 degrees and an apothem of roughly 7.46 centimeters, with the exterior angle shrinking to 30 degrees. As $n$ increases, the polygon becomes more circle-like. That trend is why regular polygons appear in numerical approximations of circles and in practical designs that need rotational symmetry without using a true curved boundary.

Regular polygons also help explain why unit discipline matters. If you enter the side length in inches, the perimeter and apothem come out in inches, and the area comes out in square inches. If you switch the side length to feet, all the outputs scale accordingly. This sounds obvious, but it is one of the easiest places to make a real mistake when sketching a project, estimating material, or checking a homework answer. The calculator handles the geometry, but you still control the meaning of the numbers you enter.

Quantity	Formula
Perimeter	$n s$
Interior Angle	$\frac{n - 2}{n} \cdot 180$ °
Exterior Angle	$\frac{360}{n}$ °
Apothem	$\frac{s}{2 \cdot \tan (\frac{π}{n})}$
Area	$\frac{1}{2} P a$

One more pattern is worth noticing. If the side length stays fixed and the side count increases, the perimeter grows linearly because you are adding more equal edges. The area grows too, but not in a purely linear way, because the apothem changes with the angle geometry. That is part of what makes regular polygons such good teaching examples: they are simple enough to compute, yet rich enough to show how geometry, trigonometry, and scaling interact.

In short, this calculator is useful both as a practical tool and as a learning aid. You can use it to check a design, verify a homework problem, compare shapes, or build intuition about how angle and side count work together. Change one input at a time, watch how the outputs move, and the formulas start to feel less like isolated rules and more like connected geometric facts.

Provide the side count and side length to see geometric properties.

Copy status will appear here.

Polygon Gate mini-game

This optional arcade mode turns the same ideas into a timing challenge. A stream of glowing regular polygons orbits the center. Your job is to fire a pulse when the polygon that matches the current target passes through the bright gate at the top of the rings. Early prompts ask for a side count. Later prompts switch to exterior and interior angles, so the game quietly trains the same relationships used by the calculator. It is short, replayable, and separate from the calculation itself.

Score
0

Time
75.0

Streak
0

Wave
1

Best
0

Polygon Gate

Click to play or press Start game. Tap, click, or press the space bar when the matching polygon crosses the glowing top gate. Correct hits build streaks and add a little time. Misses break the streak and cost time.

Quick hint: exterior angle = 360° ÷ n, so large side counts create small exterior turns.

Best score: 0

Best score is saved on this device. Use the game to build fast intuition for side count, interior angle, and exterior angle.

Regular Polygon Calculator

Understand the geometry before you calculate

Why regular polygons are worth studying

Polygon Gate mini-game

Polygon Gate

Embed this calculator

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