Regular Polygon Calculator
Calculate the area, perimeter, interior and exterior angles, apothem, and circumradius of a regular polygon from its number of sides and side length.
Cómo usar esta calculadora
- 1Enter the number of sides — 3 for a triangle, 6 for a hexagon, and so on.
- 2Enter the length of one side (all sides are equal in a regular polygon).
- 3Read the area and perimeter.
- 4See the interior and exterior angles, the apothem, and the circumradius.
Cómo funciona
Regular polygon
interior angle = (n − 2) × 180° ÷ n exterior angle = 360° ÷ n perimeter = n × side, apothem = side ÷ (2 tan(π/n)) area = ½ × perimeter × apothem = n·s² ÷ (4 tan(π/n))
A regular polygon has every side the same length and every interior angle equal. From just the number of sides and the side length, all its other measurements follow. The interior angles always sum to (n − 2) times 180 degrees — a consequence of splitting the polygon into n − 2 triangles — so each angle in a regular polygon is that sum divided by n. The exterior angles, one at each vertex, always add to a full 360 degrees, so each is 360 divided by n. The apothem is the distance from the centre to the midpoint of a side, and the area is elegantly half the perimeter times the apothem, the polygon equivalent of one-half base times height. The circumradius is the distance from the centre to a vertex, the radius of the circle that passes through all the corners. Trigonometry ties the apothem and circumradius to the side length through the angles subtended at the centre.
Ejemplo resuelto
A regular hexagon with a side length of 10 has interior angles of (6 − 2) × 180 ÷ 6 = 120°, a perimeter of 60, an apothem of about 8.66, and an area of ½ × 60 × 8.66 ≈ 259.8. Its circumradius equals its side length, 10 — a special property unique to the hexagon.
Regular Polygon Calculator: la guía completa
The angles of a regular polygon
The angles of a regular polygon follow two beautiful rules that hold no matter how many sides it has. The interior angles always sum to (n − 2) times 180 degrees. This comes from the fact that any polygon with n sides can be cut into n − 2 triangles by drawing diagonals from a single vertex, and each triangle's angles sum to 180 degrees. Since a regular polygon's angles are all equal, dividing that total by n gives each interior angle — 60 degrees for a triangle, 90 for a square, 108 for a pentagon, 120 for a hexagon, and creeping toward 180 as the sides multiply.
The exterior angles obey an even simpler rule: they always sum to exactly 360 degrees, for every polygon regardless of its number of sides. Picture walking around the perimeter of the polygon; at each corner you turn by the exterior angle, and after a complete circuit you have turned all the way around once — a full 360 degrees. In a regular polygon this total is shared equally, so each exterior angle is 360 divided by the number of sides. These two rules, interior and exterior, are among the most useful facts in elementary geometry and appear constantly in tiling, design, and construction.
Apothem, circumradius, and area
Two radii characterise a regular polygon's size. The apothem is the distance from the centre straight out to the midpoint of a side — the radius of the largest circle that fits inside the polygon, touching each side. The circumradius is the distance from the centre out to a vertex — the radius of the smallest circle that contains the polygon, passing through every corner. The polygon sits neatly between these two circles, and both radii are fixed by the side length through the angles the sides subtend at the centre.
The apothem is the key to the area, through a formula that unifies all regular polygons: area equals one-half the perimeter times the apothem. This is the polygon version of one-half base times height, and it works because a regular polygon can be sliced into n identical triangles meeting at the centre, each with base equal to a side and height equal to the apothem. Summing their areas gives one-half times the total base (the perimeter) times the height (the apothem). This single elegant formula handles a triangle, a hexagon, or a hundred-sided polygon equally well, which is why the apothem is such a central quantity.
Toward the circle, and where polygons appear
As a regular polygon gains more sides, it looks more and more like a circle. A hexagon is noticeably angular, a dodecagon less so, and a hundred-sided polygon is nearly indistinguishable from a circle to the eye. This is not just an appearance: the polygon's area approaches the circle's area, its perimeter approaches the circumference, and its apothem and circumradius converge on the circle's radius. Ancient mathematicians exploited exactly this, approximating a circle by polygons with ever more sides to estimate π — Archimedes famously bounded π using 96-sided polygons.
Regular polygons are everywhere in the human and natural world because their symmetry is both efficient and beautiful. The hexagon in particular tiles the plane perfectly with no gaps and encloses area efficiently, which is why honeybees build hexagonal cells and why hexagonal patterns appear in everything from bolt heads to graphene. Squares and equilateral triangles also tile perfectly, making them staples of flooring, structures, and design. From the pentagon of a soccer ball to the octagon of a stop sign, regular polygons are chosen for their balance, and the measurements this calculator provides — angles, area, and radii — are what designers and builders need to work with them.
Preguntas frecuentes
How do I find the area of a regular polygon?
Multiply half the perimeter by the apothem (the distance from the centre to the middle of a side): area = ½ × perimeter × apothem. Equivalently, area = n × s² ÷ (4 tan(π/n)). A hexagon with side 10 has an area of about 259.8.
What is the interior angle of a regular polygon?
Each interior angle is (n − 2) × 180° ÷ n, where n is the number of sides. So a hexagon's interior angles are (6 − 2) × 180 ÷ 6 = 120°. The angles increase toward 180° as the polygon gains more sides and approaches a circle.
What is the apothem?
The apothem is the distance from the centre of a regular polygon to the midpoint of any side — the radius of the inscribed circle that touches each side. It equals side ÷ (2 tan(π/n)) and is the key to the area formula, ½ × perimeter × apothem.
Why do exterior angles always add up to 360°?
Because walking around the polygon's perimeter, you turn by each exterior angle and complete one full rotation — 360° — after going all the way around. This holds for any polygon. In a regular one, the angles are equal, so each exterior angle is 360° ÷ n.