Calculat.dev LogoCalculat.dev - All Calculators, One Place.devAll Calculators, One Place.
Math CalculatorsPopular Tool Runs Locally

Regular Polygon Calculator

Calculate regular n-sided polygon area, perimeter, apothem, circumradius, interior angles, and exterior angles with step-by-step steps.

Regular 8-gon Properties
Area
120.7107
Perimeter
40
Apothem (r)
6.0355
Interior Angle
135°
Deduction Steps:
Regular polygon with n = 8 sides, side length s = 5
Perimeter: P = n × s = 8 × 5 = 40
Interior Angle: ((n - 2) × 180°) / n = ((8 - 2) × 180) / 8 = 135°
Exterior Angle: 360° / n = 360 / 8 = 45°
Apothem (inradius): a = s / (2 · tan(π/n)) = 6.0355
Circumradius: R = s / (2 · sin(π/n)) = 6.5328
Area: A = ½ × P × a = 0.5 × 40 × 6.0355 = 120.7107
🎓

Did this calculation save you time?

Student Project

Hi! I'm a student developer building Calculat in my spare time. I was tired of searching for basic math tools and having to click through 10 spammy popups, loan ads, and cookie trackers.

I keep this website 100% free, private, and ad-free. If this helped you with your homework, project, or finances today, bookmarking this page or telling a friend helps me keep building more free tools!

Press ⌘ + D to Bookmark
Suggest What I Build Next

What Is a Regular Polygon Calculator?

A regular polygon is an n-sided polygon that is both equiangular (all angles are equal) and equilateral (all sides have equal length). As the number of sides approaches infinity, the regular polygon approaches a circle.

How to Use This Calculator

  1. Enter the number of sides (n ≥ 3) and side length (s).
  2. View the calculated Area, Perimeter, Apothem, Circumradius, and Angles.
  3. Review the generalized trigonometric formulas.

Regular n-gon Geometric Formulas

A = \frac{n s^2}{4\tan\left(\frac{\pi}{n}\right)}, \quad a = \frac{s}{2\tan\left(\frac{\pi}{n}\right)}, \quad \theta_{\text{int}} = \frac{(n-2)\times 180^\circ}{n}

Area is calculated using trigonometry by dividing the n-gon into n isosceles triangles. Interior angle uses the polygon angle sum theorem.

Worked Example

Scenario: Calculate regular octagon (n = 8) with side length s = 5 cm

Perimeter: P = 8 × 5 = 40 cm

Interior angle: (8 - 2) × 180° / 8 = 1080° / 8 = 135°

Apothem: a = 5 / (2 × tan(180° / 8)) = 5 / (2 × tan(22.5°)) ≈ 6.0355 cm

Area: A = ½ × 40 × 6.0355 ≈ 120.7107 cm²

Result: Area ≈ 120.71 cm², Perimeter = 40 cm, Interior Angle = 135°

Tips & Key Notes

  • Exterior angles of any convex polygon always sum to 360°; for an n-gon, each exterior angle is 360° / n.
  • The interior and exterior angles at any vertex are supplementary: θ_int + θ_ext = 180°.
  • The circumradius is R = s / (2 × sin(π/n)).

Frequently Asked Questions

What is the smallest number of sides a polygon can have?

A polygon must have at least 3 sides (a triangle).

What is the difference between inradius and circumradius of a polygon?

The inradius (apothem) is the radius of the inscribed circle touching the sides; the circumradius is the radius of the circle passing through the vertices.

What is an apothem?

The apothem is a line segment from the center of a regular polygon perpendicular to one of its sides.

How do you find the total sum of angles in an n-gon?

The sum of interior angles is always (n - 2) × 180 degrees.

Related Calculators

Explore similar tools
PopularMath

Pentagon Calculator

Calculate regular pentagon area, perimeter, apothem, diagonals, and interior angles from side length with step-by-step math.

Calculate
PopularMath

Hexagon Calculator

Calculate regular hexagon area, perimeter, apothem, long diagonal, short diagonal, and circumradius with step-by-step steps.

Calculate
PopularMath

Circle Calculator

Calculate circle area, circumference, diameter, radius, arc length, and sector area with step-by-step geometric derivations.

Calculate