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30
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sq units
120
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A regular polygon is one of the most fundamental objects in Euclidean geometry, defined as a closed plane figure composed of a finite number of equal-length straight sides that meet at equal interior angles. From the equilateral triangle with three sides to the regular myriagon with ten thousand, regular polygons form an infinite family of shapes that converge toward the circle as the number of sides grows without bound. The study of regular polygons dates to antiquity: Greek mathematicians including Euclid devoted substantial portions of the Elements to their construction, and the question of which regular polygons can be constructed with compass and straightedge alone remained open until Carl Friedrich Gauss resolved it in 1796 by constructing the regular 17-gon.
The Regular Polygon Calculator provides a comprehensive analysis of any regular polygon given just two parameters—the number of sides and the length of each side. It computes the total area enclosed by the polygon, the perimeter (the sum of all side lengths), the apothem (the perpendicular distance from the center to the midpoint of any side), and both the interior and exterior angles. These measurements are essential in architecture, tiling design, engineering, computer graphics, and CNC machining, where regular polygons appear in floor plans, gear teeth, bolt patterns, and tessellation algorithms.
The area of a regular polygon is derived by decomposing it into n congruent isosceles triangles, each with a base equal to the side length and a height equal to the apothem. Summing the areas of these triangles yields the elegant formula A = (n · s²) / (4 tan(π/n)). The perimeter is simply P = n · s, while the apothem follows as a = s / (2 tan(π/n)). Interior angles are determined by the formula (n − 2) · 180° / n, and each exterior angle is its supplement, equal to 360° / n.
Understanding these relationships is crucial in practical applications. In architecture, regular polygons determine the geometry of towers, rotundas, and pavilions—the Pentagon building in Arlington, Virginia, is literally a regular pentagon. In mechanical engineering, hexagonal bolt heads and nuts rely on the properties of the regular hexagon for optimal wrench engagement. In computer graphics, regular polygons serve as base meshes for approximating circles and curved surfaces, and their symmetry properties simplify rendering calculations through rotational matrices.
The apothem deserves special attention because it connects the polygon to its inscribed circle—the largest circle that fits entirely within the polygon touches each side at its midpoint, and the radius of this inscribed circle equals the apothem. Similarly, the circumradius—the radius of the circumscribed circle passing through every vertex—is given by R = s / (2 sin(π/n)). Together, the apothem and circumradius characterize the two canonical circles associated with every regular polygon.
As the number of sides increases, the regular polygon increasingly resembles a circle. The ratio of the polygon's area to the area of its circumscribed circle approaches unity, and the perimeter approaches the circumference of that circle. This limiting behavior provides one of the oldest methods for approximating π: Archimedes bounded π between the perimeters of inscribed and circumscribed regular 96-gons, obtaining the famous estimate 3.1408 < π < 3.1429. This calculator lets you explore this convergence numerically by entering large values of n and observing how the computed quantities approach their circular limits.
Whether you are designing a gazebo footprint, calculating material for a stop-sign blank, programming a game engine’s collision geometry, or simply studying polygon properties for a mathematics course, the Regular Polygon Calculator delivers precise results instantly.
The calculator uses the following formulas for a regular polygon with n sides of length s:
$$A = \frac{n \cdot s^2}{4 \tan\left(\frac{\pi}{n}\right)}$$
$$P = n \cdot s$$
$$a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)}$$
$$\theta_{\text{interior}} = \frac{(n-2) \cdot 180°}{n}$$
$$\theta_{\text{exterior}} = \frac{360°}{n}$$
The area formula is derived by splitting the polygon into n isosceles triangles radiating from the center, each with base s and height equal to the apothem a. The area of one triangle is (1/2) · s · a, so the total area is n · (1/2) · s · a = (n · s · a) / 2. Substituting the apothem expression gives the closed-form result above.
The area tells you the total region enclosed by the polygon, useful for material estimation and spatial planning. The perimeter gives the total boundary length, important for fencing, edging, or border calculations. The apothem is the shortest distance from the center to any side and equals the radius of the inscribed circle. The interior angle determines how the polygon's corners fit together—critical for tiling and tessellation. The exterior angle is the supplement of the interior angle and always sums to 360° across all vertices regardless of the number of sides.
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A regular hexagon with side 10 has area ≈ 259.81 square units, matching (3√3/2)·10².
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A regular pentagon with side 7 has area ≈ 84.30 square units and interior angles of 108°.
A regular polygon is a polygon where all sides have equal length and all interior angles are equal. Examples include the equilateral triangle (3 sides), square (4 sides), regular pentagon (5 sides), and regular hexagon (6 sides). The regularity condition means the shape has maximum symmetry—it is both equilateral (equal sides) and equiangular (equal angles).
The apothem is the perpendicular distance from the center of a regular polygon to the midpoint of any of its sides. It equals the radius of the inscribed circle (incircle) and is central to the area formula: A = (1/2) · perimeter · apothem. The apothem is widely used in engineering to determine clearance distances for polygonal cross-sections.
The polygon is divided into n congruent isosceles triangles, each with base s and height equal to the apothem. The total area is then n times the area of one triangle: A = n · (1/2) · s · a = ns² / (4 tan(π/n)). This approach works for any regular polygon regardless of the number of sides.
When you walk along the perimeter of any convex polygon and turn at each vertex by the exterior angle, you complete exactly one full rotation (360°) by the time you return to your starting point. For a regular polygon with n sides, each exterior angle is 360°/n, and n such turns sum to 360°.
Yes. As n increases, the regular polygon increasingly approximates a circle. For very large n, the area approaches πR² (where R is the circumradius) and the perimeter approaches 2πR. You can explore this convergence by trying n = 100, 1000, or more.
The apothem is the distance from the center to the midpoint of a side (inscribed circle radius), while the circumradius is the distance from the center to a vertex (circumscribed circle radius). They are related by: apothem = circumradius · cos(π/n). The circumradius is always larger than the apothem, and the difference decreases as n increases.
Roboculator Team
The Roboculator Team explains calculations, planning tools, and practical formulas in clear language for real-life situations.
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