What is the interior angle of a regular polygon?

Interior angle = (n-2) x 180 / n degrees. Triangle: 60 degrees, square: 90 degrees, pentagon: 108 degrees, hexagon: 120 degrees, octagon: 135 degrees. The sum of all interior angles is (n-2) x 180 degrees.

What is the circumradius of a regular polygon?

The circumradius R is the radius of the circumscribed circle (the circle that passes through all vertices). R = s / (2 x sin(pi/n)) where s is the side length and n is the number of sides.

What is the apothem of a regular polygon?

The apothem (inradius) is the perpendicular distance from the center to the midpoint of any side. Apothem = s / (2 x tan(pi/n)) = R x cos(pi/n). It is the radius of the inscribed circle.

How do I find the side length from the circumradius?

Use side length s = 2 x R x sin(pi/n), where R is the circumradius and n is the number of sides. For example, a regular hexagon with R = 10 has s = 2 x 10 x sin(pi/6) = 2 x 10 x 0.5 = 10.

Polygon Calculator

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

Area = (n x s^2 x cot(pi/n)) / 4, where n is the number of sides and s is the side length. Equivalently, Area = (perimeter x apothem) / 2 = n x s x a / 2 where a is the apothem.

Q: How many diagonals does a polygon have?

A polygon with n sides has n(n-3)/2 diagonals. A triangle has 0, a square has 2, a pentagon has 5, a hexagon has 9, an octagon has 20.

Q: What is a regular polygon?

A regular polygon has all sides equal in length and all interior angles equal. Examples include the equilateral triangle (3 sides), square (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), and decagon (10).

Q: What is the exterior angle of a regular polygon?

The exterior angle is 360 / n degrees. The exterior angle is always supplementary to the interior angle: exterior angle + interior angle = 180 degrees. The sum of all exterior angles of any convex polygon is always 360 degrees.

Enter the number of sides and side length (or circumradius) to find area, perimeter, angles, apothem, and diagonals for any regular polygon.

🔷 What is a Regular Polygon?

A regular polygon is a flat, closed shape with all sides equal in length and all interior angles equal in measure. The simplest regular polygon is the equilateral triangle (3 sides). Other common examples include the square (4 sides), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), and decagon (10). This calculator supports regular polygons with 3 to 20 sides.

Regular polygons appear throughout architecture, nature, and engineering. The hexagonal tiles on bathroom floors, the cross-section of a bolt head (hexagon), the shape of a stop sign (octagon), honeycomb cells (hexagons again), and the Pentagon building in Washington D.C. are all regular or near-regular polygons. In mathematics, regular polygons are used to approximate the area of a circle (as the number of sides approaches infinity, the area approaches pi times r squared).

A key distinction is between regular and irregular polygons. An irregular polygon has sides or angles that differ from each other. A rectangle, for example, is an irregular polygon (unless it is a square). This calculator is specifically for regular polygons where the high degree of symmetry allows simple, exact formulas. For irregular polygons, you would need to specify all side lengths and angles separately.

The properties of a regular polygon are fully determined by just two values: the number of sides n and one measurement (such as the side length or circumradius). Once these two values are known, every other property follows from exact trigonometric formulas. This is what makes regular polygons so mathematically elegant and practically useful.

📐 Formulas

Area = (n × s²) ÷ (4 × tan(π ÷ n))

n = number of sides

s = side length

Perimeter = n × s

Interior angle = (n − 2) × 180 ÷ n (degrees)

Exterior angle = 360 ÷ n (degrees)

Circumradius R = s ÷ (2 × sin(π ÷ n))

Apothem r = s ÷ (2 × tan(π ÷ n)) = R × cos(π ÷ n)

Number of diagonals = n × (n − 3) ÷ 2

Example: Regular hexagon, s = 5: Area = (6 × 25) / (4 × tan(30°)) = 150 / 2.309 = 64.95 sq units

📖 How to Use This Calculator

Steps

Choose your input mode - Select From Side Length if you know each side's length. Select From Circumradius if you know the radius of the circle that passes through all vertices of the polygon.

Set the number of sides - Drag the slider or type a number from 3 to 20. The polygon name (triangle, hexagon, octagon, etc.) is displayed in the result so you always know which shape you are computing.

Enter the measurement - Type the side length (or circumradius) in any consistent unit. The same unit is used for all length outputs. Area is in square units.

Read all properties - The results show area, perimeter, interior angle, exterior angle, circumradius R, apothem (inradius r), and the number of diagonals, all computed instantly.

💡 Example Calculations

Example 1 - Regular Hexagon with Side 6 cm

Find all properties of a regular hexagon with side length 6 cm.

n = 6, s = 6 cm.

Area = (6 x 36) / (4 x tan(30 deg)) = 216 / (4 x 0.5774) = 216 / 2.3094 = 93.53 cm2.

Perimeter = 6 x 6 = 36 cm. Interior angle = (6-2) x 180 / 6 = 120 deg. Exterior angle = 60 deg.

Circumradius R = 6 / (2 x sin(30 deg)) = 6 / 1 = 6 cm (for a hexagon, R = s). Apothem = 6 x cos(30 deg) = 5.196 cm. Diagonals = 6 x 3 / 2 = 9.

Area = 93.53 cm2, Perimeter = 36 cm, Interior angle = 120 deg

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Example 2 - Regular Octagon with Side 4 m

Find the area and angles of a regular stop-sign octagon with side 4 m.

n = 8, s = 4 m.

Area = (8 x 16) / (4 x tan(22.5 deg)) = 128 / (4 x 0.4142) = 128 / 1.6569 = 77.25 m2.

Interior angle = (8-2) x 180 / 8 = 135 deg. Exterior angle = 45 deg.

Circumradius = 4 / (2 x sin(22.5 deg)) = 4 / 0.7654 = 5.226 m. Diagonals = 8 x 5 / 2 = 20.

Area = 77.25 m2, Interior angle = 135 deg, Diagonals = 20

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Example 3 - Pentagon Inscribed in a Circle of Radius 10 cm

A regular pentagon is inscribed in a circle with radius 10 cm. Find its side length and area.

Use Circumradius mode. n = 5, R = 10 cm.

Side length s = 2 x 10 x sin(pi/5) = 20 x sin(36 deg) = 20 x 0.5878 = 11.756 cm.

Area = (5 x 11.756^2) / (4 x tan(36 deg)) = (5 x 138.20) / (4 x 0.7265) = 691.0 / 2.906 = 237.76 cm2.

Interior angle = (5-2) x 180 / 5 = 108 deg. Diagonals = 5 x 2 / 2 = 5.

Side = 11.756 cm, Area = 237.76 cm2, Interior angle = 108 deg

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❓ Frequently Asked Questions

What is the formula for the area of a regular polygon?+

Area = (n x s^2) / (4 x tan(pi/n)), where n is the number of sides and s is the side length. Equivalently, Area = (perimeter x apothem) / 2 = n x s x a / 2, where a is the apothem. For a hexagon with s = 1: Area = (6 x 1) / (4 x tan(30 deg)) = 6 / 2.309 = 2.598 square units.

How do I find the interior angle of a regular polygon?+

Interior angle = (n-2) x 180 / n degrees. The sum of all interior angles of a polygon is (n-2) x 180 degrees. For a regular polygon, this sum is divided equally among all n angles. Key values: triangle 60 deg, square 90 deg, pentagon 108 deg, hexagon 120 deg, octagon 135 deg, decagon 144 deg.

What is the circumradius of a polygon?+

The circumradius R is the radius of the circle that passes through all the vertices of the polygon (the circumscribed circle). R = s / (2 x sin(pi/n)). For a regular hexagon, R equals the side length s exactly, since sin(pi/6) = 0.5. For a square with s = 10, R = 10 / (2 x sin(45 deg)) = 10 / 1.414 = 7.071.

What is the apothem (inradius) of a polygon?+

The apothem r is the perpendicular distance from the center of the polygon to the midpoint of any side. It is the radius of the inscribed circle. r = s / (2 x tan(pi/n)) = R x cos(pi/n). Using the apothem, Area = n x s x r / 2 = perimeter x r / 2, which is a convenient alternative area formula.

How many diagonals does a polygon have?+

Number of diagonals = n(n-3)/2. A triangle (n=3) has 0 diagonals. A square has 2. A pentagon has 5. A hexagon has 9. An octagon has 20. A decagon has 35. The formula comes from the fact that from each of the n vertices, you can draw n-3 diagonals (not to itself and not to its two adjacent vertices), giving n(n-3) total, divided by 2 to avoid double counting.

What is the exterior angle of a regular polygon?+

Exterior angle = 360 / n degrees. The exterior angle is formed between one side of the polygon and the extension of the adjacent side. The sum of all exterior angles of any convex polygon is always 360 degrees, which is why each exterior angle of a regular polygon is 360/n.

What is the difference between a regular and irregular polygon?+

A regular polygon has all sides equal and all angles equal. A rectangle, rhombus, or general quadrilateral is an irregular polygon even if it has some equal sides or angles. Only the square is a regular quadrilateral. This calculator applies only to regular polygons; for irregular polygons, side lengths and angles must be specified individually.

What happens to a polygon as the number of sides increases?+

As n increases, a regular polygon increasingly approximates a circle. The area approaches pi x r^2 and the perimeter approaches 2 x pi x r (where r is the circumradius). Archimedes used 96-sided polygons to approximate pi to within 0.002%. The interior angle approaches 180 degrees, and the exterior angle approaches 0 degrees. A polygon with infinitely many sides is mathematically equivalent to a circle.

How do I find the side length from the area of a regular polygon?+

Rearrange the area formula: s = sqrt(4 x Area x tan(pi/n) / n). For example, if a regular hexagon has area 100 cm2: s = sqrt(4 x 100 x tan(30 deg) / 6) = sqrt(400 x 0.5774 / 6) = sqrt(38.49) = 6.204 cm. This calculator solves the forward problem; for the inverse, use this rearranged formula directly.

What are the names of regular polygons with 3 to 12 sides?+

3 sides: equilateral triangle. 4 sides: square. 5 sides: regular pentagon. 6 sides: regular hexagon. 7 sides: regular heptagon (or septagon). 8 sides: regular octagon. 9 sides: regular nonagon (or enneagon). 10 sides: regular decagon. 11 sides: regular hendecagon (or undecagon). 12 sides: regular dodecagon.

Why does a regular hexagon have circumradius equal to side length?+

For a regular hexagon (n = 6), the circumradius formula gives R = s / (2 x sin(pi/6)) = s / (2 x 0.5) = s. So R = s exactly. This is because a regular hexagon can be divided into 6 equilateral triangles, each with side equal to R. This unique property means a hexagonal tile grid perfectly tiles around any point, which is why hexagonal patterns appear in nature (honeycomb) and engineering (carbon structures).

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📌 Quick Tips

💡A regular polygon with n sides has interior angle = (n-2) x 180 / n degrees. For a hexagon: (6-2) x 180 / 6 = 120 degrees.

💡The number of diagonals in any polygon is n(n-3)/2. A pentagon has 5 x 2 / 2 = 5 diagonals.

💡The apothem is the perpendicular distance from the center to any side. Area = (perimeter x apothem) / 2.

💡For a regular hexagon with side s, the circumradius R equals the side length s exactly.

💡Use the Circumradius mode when you know the circle that the polygon is inscribed in.