Regular Polygon Calculator
Area
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Perimeter
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Interior Angle
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Apothem
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How Regular Polygons Work
A regular polygon is a two-dimensional shape with all sides of equal length and all interior angles of equal measure. The study of regular polygons dates back to ancient Greek mathematicians, particularly Euclid's Elements (circa 300 BC), which included constructions for regular polygons with 3, 4, 5, 6, and 15 sides. The mathematical properties of regular polygons form the foundation of geometry, architecture, and engineering design. Regular polygons appear throughout nature (hexagonal honeycombs, pentagonal starfish), architecture (octagonal stop signs, hexagonal floor tiles), and manufacturing (hexagonal bolt heads, pentagonal soccer ball panels).
This calculator computes the area, perimeter, interior angle, and apothem for any regular polygon given the number of sides and side length. As the number of sides increases, a regular polygon increasingly approximates a circle, with the area approaching the formula pi * r^2. Archimedes used this principle to estimate pi by inscribing and circumscribing 96-sided polygons around a circle, obtaining an approximation accurate to two decimal places. Our area calculator handles additional shapes.
Regular Polygon Formulas
For a regular polygon with n sides and side length s, the key formulas are:
Interior Angle = (n - 2) * 180 / n degrees
Exterior Angle = 360 / n degrees
Apothem (a) = s / (2 * tan(pi/n))
Area = (1/2) * n * s * a = (1/2) * perimeter * apothem
Circumradius (R) = s / (2 * sin(pi/n))
Worked example: For a regular hexagon (n = 6) with side length 5. Interior angle = (6-2)*180/6 = 120 degrees. Apothem = 5 / (2 * tan(pi/6)) = 5 / (2 * 0.5774) = 4.33. Area = (1/2) * 6 * 5 * 4.33 = 64.95 square units. Perimeter = 6 * 5 = 30. Circumradius = 5 / (2 * sin(pi/6)) = 5. For woodworking miter cuts on polygons, try the miter angle calculator.
Key Terms You Should Know
Regular Polygon is a polygon where all sides are equal in length and all interior angles are equal in measure. This distinguishes it from irregular polygons where sides and angles may vary.
Apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It equals the radius of the inscribed circle (incircle).
Circumradius is the distance from the center to any vertex. It equals the radius of the circumscribed circle (circumcircle) that passes through all vertices.
Interior Angle is the angle formed inside the polygon at each vertex. The sum of all interior angles is (n - 2) * 180 degrees.
Tessellation is a pattern of shapes that covers a plane with no gaps or overlaps. Only three regular polygons can tessellate by themselves: triangles, squares, and hexagons.
Properties of Common Regular Polygons
The table below summarizes the key properties of regular polygons from 3 to 12 sides, with side length normalized to 1 for easy comparison.
| Name | Sides | Interior Angle | Area (s=1) | Apothem (s=1) | Tessellates? |
|---|---|---|---|---|---|
| Equilateral Triangle | 3 | 60.00 | 0.433 | 0.289 | Yes |
| Square | 4 | 90.00 | 1.000 | 0.500 | Yes |
| Pentagon | 5 | 108.00 | 1.720 | 0.688 | No |
| Hexagon | 6 | 120.00 | 2.598 | 0.866 | Yes |
| Heptagon | 7 | 128.57 | 3.634 | 1.038 | No |
| Octagon | 8 | 135.00 | 4.828 | 1.207 | No |
| Decagon | 10 | 144.00 | 7.694 | 1.539 | No |
| Dodecagon | 12 | 150.00 | 11.196 | 1.866 | No |
Practical Examples
Example 1: Stop sign dimensions. A standard U.S. stop sign is a regular octagon (8 sides) with a side length of approximately 12.4 inches. Interior angle = (8-2)*180/8 = 135 degrees. Apothem = 12.4 / (2 * tan(pi/8)) = 12.4 / 0.8284 = 14.97 inches. Area = (1/2) * 8 * 12.4 * 14.97 = 743.2 square inches.
Example 2: Honeycomb hexagons. A typical honeycomb cell has a side length of about 3 mm. For this regular hexagon: apothem = 3 / (2 * tan(30)) = 2.60 mm, area = (1/2) * 6 * 3 * 2.60 = 23.4 square mm. The hexagonal shape is optimal because it maximizes area while minimizing perimeter, making it the most material-efficient shape for tiling a surface. This principle is used in aerospace engineering for honeycomb sandwich panels.
Example 3: Pentagon-shaped garden bed. A gardener wants to build a regular pentagonal raised bed with 4-foot sides. Interior angle = (5-2)*180/5 = 108 degrees. The miter cut angle for each corner board would be 108/2 = 54 degrees. Area = (1/2) * 5 * 4 * 2.75 = 27.5 square feet, providing ample growing space for a variety of plants. Use our trapezoid calculator for non-regular garden bed shapes.
Tips and Strategies
- Use the apothem for area calculations. The formula A = (1/2) * perimeter * apothem is the most practical way to compute area for any regular polygon and is easier to remember than side-specific formulas.
- For construction projects, calculate miter angles. When building polygonal frames, each miter cut angle is (180 - interior angle) / 2 degrees. For an octagon, this is (180 - 135) / 2 = 22.5 degrees.
- Approximate circles with high-sided polygons. A 36-sided polygon has interior angles of 170 degrees and an area that is 99.6% of the circumscribed circle. This technique is used in CNC machining and 3D printing.
- Remember the tessellation rule. For a regular polygon to tessellate alone, its interior angle must divide evenly into 360. Only 60 (triangle), 90 (square), and 120 (hexagon) satisfy this condition.
- Convert between apothem and circumradius. If you know the circumradius R, the side length s = 2 * R * sin(pi/n) and the apothem a = R * cos(pi/n). This is useful when working with inscribed or circumscribed circles.
Frequently Asked Questions
What is the formula for the interior angle of a regular polygon?
The interior angle of a regular polygon is (n - 2) * 180 / n degrees, where n is the number of sides. For a triangle (n=3), each interior angle is 60 degrees. For a square (n=4), it is 90 degrees. For a regular pentagon (n=5), 108 degrees. For a regular hexagon (n=6), 120 degrees. For a regular octagon (n=8), 135 degrees. As the number of sides increases, the interior angle approaches but never reaches 180 degrees.
What is the apothem of a regular polygon?
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any side. It is calculated using the formula a = s / (2 * tan(pi/n)), where s is the side length and n is the number of sides. The apothem is also the radius of the largest circle that fits entirely inside the polygon (the inscribed circle). For a regular hexagon with side length 5, the apothem is 5 / (2 * tan(pi/6)) = 5 / (2 * 0.5774) = 4.33.
What is the exterior angle of a regular polygon?
The exterior angle of a regular polygon is 360 / n degrees, where n is the number of sides. For a regular hexagon, each exterior angle is 360 / 6 = 60 degrees. For an octagon, it is 360 / 8 = 45 degrees. The sum of all exterior angles of any convex polygon is always exactly 360 degrees, regardless of the number of sides. Each exterior angle is supplementary to its corresponding interior angle, meaning they add up to 180 degrees.
Which regular polygons can tile a flat plane?
Only three regular polygons can tile a flat plane by themselves: equilateral triangles (interior angle 60 degrees, six fit around a point), squares (90 degrees, four fit around a point), and regular hexagons (120 degrees, three fit around a point). These are the only regular polygons whose interior angles divide evenly into 360 degrees. This was proven by ancient Greek mathematicians and the result still holds. Semi-regular tilings using combinations of different regular polygons are also possible.
How do you calculate the area of a regular polygon?
The area of a regular polygon is calculated using the formula A = (1/2) * n * s * a, where n is the number of sides, s is the side length, and a is the apothem. Equivalently, since the perimeter P = n * s, the formula can be written as A = (1/2) * P * a. For a regular hexagon with side length 10, the apothem is 8.66 and the area is (1/2) * 6 * 10 * 8.66 = 259.8 square units. As the number of sides increases, the area formula approaches pi * r^2, the formula for a circle.
What is the difference between the apothem and the circumradius?
The apothem is the distance from the center to the midpoint of a side (the inradius), while the circumradius is the distance from the center to a vertex. The circumradius R = s / (2 * sin(pi/n)) and the apothem a = s / (2 * tan(pi/n)). The circumradius is always larger than the apothem. For a regular hexagon with side length 5, the circumradius equals the side length (5), while the apothem is 4.33. The ratio between them depends on the number of sides and approaches 1 as the polygon approaches a circle.