Regular Polygon Calculator
Calculate area, perimeter, apothem, and radius with multiple input options
Input Values
Results
Step-by-Step Solution
r = s / (2 × sin(π/n))
r = 10 / (2 × sin(π/6))
r = 0.0000
a = s / (2 × tan(π/n))
a = 10 / (2 × tan(π/6))
a = 0.0000
P = n × s
P = 6 × 0.0000
P = 0.0000
A = (1/2) × P × a
A = (1/2) × 0.0000 × 0.0000
A = 0.0000
Understanding Regular Polygons
What is a Regular Polygon?
A regular polygon is a polygon with all sides of equal length and all interior angles equal. Regular polygons are both equilateral (equal sides) and equiangular (equal angles). They possess rotational symmetry and can be inscribed in a circle.
Key Measurements
Radius (Circumradius): The distance from the center to any vertex. r = s / (2 × sin(π/n))
Apothem (Inradius): The perpendicular distance from the center to the midpoint of any side. a = s / (2 × tan(π/n))
Relationship: a = r × cos(π/n)
Area Formula: A = (1/2) × perimeter × apothem = (n × s²) / (4 × tan(π/n))
Derivation of Area Formula
A regular polygon can be divided into n congruent isosceles triangles, each with:
- Base = side length (s)
- Height = apothem (a)
- Area of one triangle = (1/2) × s × a
- Total area = n × (1/2) × s × a = (1/2) × (n × s) × a = (1/2) × perimeter × apothem
Properties of Regular Polygons
- All vertices lie on a circle (circumcircle) with radius r
- All sides are tangent to a circle (incircle) with radius a (apothem)
- Central angle = 360°/n (angle subtended at center by one side)
- Interior angle = (n-2) × 180°/n
- Exterior angle = 360°/n
- Has n lines of symmetry and rotational symmetry of order n
Frequently Asked Questions
What is the difference between radius and apothem?
The radius (circumradius) extends from the center to a vertex, while the apothem (inradius) is the perpendicular distance from the center to the midpoint of a side. The apothem is always shorter than the radius, and they are related by a = r × cos(π/n).
How do you find the area if you only know the radius?
First calculate the side length: s = 2r × sin(π/n), then the apothem: a = r × cos(π/n), and finally the area: A = (1/2) × n × s × a. Alternatively, use A = (n × r² × sin(2π/n)) / 2.
Why is the apothem important?
The apothem is crucial for calculating the area of a regular polygon. The standard area formula A = (1/2) × perimeter × apothem relies on the apothem. It also represents the radius of the largest circle that can fit inside the polygon (incircle).
What happens to a regular polygon as the number of sides increases?
As the number of sides increases, a regular polygon approaches the shape of a circle. The interior angles increase toward 180°, and the ratio of apothem to radius approaches 1. This concept was historically used to approximate π.
Can you construct all regular polygons with compass and straightedge?
No, only certain regular polygons can be constructed with compass and straightedge. These include polygons with 3, 4, 5, 6, 8, 10, 12, 15, 16, 17, 20, 24, 32, 34, 40, 48, 51, 60, 64, 68, 80, 85, 96 sides (and some others). A regular heptagon (7 sides) cannot be constructed this way.
What is the central angle of a regular polygon?
The central angle is the angle formed at the center of the polygon by lines drawn to two consecutive vertices. It equals 360°/n and is also equal to the exterior angle of the polygon.
How are regular polygons used in tessellations?
Only three regular polygons can tessellate (tile) a plane by themselves: equilateral triangles, squares, and regular hexagons. This is because their interior angles (60°, 90°, and 120°) divide evenly into 360°. However, combinations of different regular polygons can create semi-regular tessellations.
What is the relationship between perimeter and area?
For regular polygons with the same perimeter, those with more sides have greater area. A circle (infinite-sided polygon) has the maximum area for a given perimeter. This is related to the isoperimetric problem in mathematics.