Star Shape Calculator

Determine the properties of a regular star polygon.

Enter the parameters of the star below to calculate its area, perimeter, and interior angle.

Practical Examples

Click on an example to load its data into the calculator.

Classic 5-Point Star (Pentagram)

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A standard pentagram, often seen in flags and symbols.

Points (n): 5

Outer Radius (R): 10

Inner Radius (r): 3.8

6-Point Star (Hexagram)

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A hexagram, also known as the Star of David.

Points (n): 6

Outer Radius (R): 10

Inner Radius (r): 5.8

Sharp 7-Point Star

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A heptagram with a small inner radius, creating sharp points.

Points (n): 7

Outer Radius (R): 12

Inner Radius (r): 4

Full 8-Point Star

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An octagram with a large inner radius, making it appear fuller.

Points (n): 8

Outer Radius (R): 10

Inner Radius (r): 7

Other Titles
Understanding the Star Shape Calculator: A Comprehensive Guide
An in-depth look at the geometry of star polygons, their properties, and how to calculate them.

What is a Star Polygon?

  • Definition of a Star Polygon
  • Key Parameters: Points and Radii
  • Visualizing the Shape
A star polygon is a type of non-convex polygon. Regular star polygons are created by connecting vertices of a simple, regular n-sided polygon. The shape is defined by the number of points (n), an outer radius (R) that touches the tips of the star, and an inner radius (r) that touches the inner 'valley' vertices.
Key Parameters
Number of Points (n): This determines the fundamental shape of the star (e.g., n=5 for a pentagram). For a valid star, n must be 5 or greater. Outer Radius (R): The distance from the center to any outer tip. Inner Radius (r): The distance from the center to any inner vertex. For a valid star, r must be less than R.

Common Star Types

  • Pentagram (5 points)
  • Hexagram (6 points)
  • Heptagram (7 points)

Step-by-Step Guide to Using the Calculator

  • Entering the Number of Points
  • Defining the Radii
  • Interpreting the Results
Using the Star Shape Calculator is straightforward. Follow these steps to get the geometric properties of your star.
1. Enter Number of Points (n)
Input the desired number of points for your star. This must be a whole number and at least 5.
2. Enter Outer and Inner Radii (R and r)
Provide the lengths for the outer and inner radii. Both must be positive numbers, and the inner radius (r) must be smaller than the outer radius (R) to form a valid star shape.
3. Calculate and Analyze
Click the 'Calculate' button. The tool will instantly provide the star's total area, perimeter, and the angle at each of its points.

Input Examples

  • n=5, R=10, r=4
  • n=8, R=20, r=15

Mathematical Formulas Behind the Calculations

  • Calculating the Area
  • Calculating the Perimeter
  • Calculating the Interior Angle
The calculator uses standard geometric formulas to determine the properties of a regular star polygon from its radii and point count.
Area Formula
The area of the star is calculated by treating it as a series of 2n triangles with their vertices at the center, an outer point, and an adjacent inner point. The formula is: Area = n × R × r × sin(π / n).
Perimeter Formula
The perimeter is the total length of the star's outer boundary. It consists of 2n equal segments connecting the inner and outer vertices. The length of one segment is found using the Law of Cosines: l = √(R² + r² - 2Rr cos(π/n)). The total perimeter is 2n × l.
Interior Angle Formula
The angle at each outer point of the star is also calculated using the Law of Cosines on the triangle formed by an outer point and its two adjacent inner vertices. The formula for the angle (α) is: α = arccos(1 - (2r sin(π/n))² / (2l²)), where l is the segment length.

Formula Components

  • n = Number of points
  • R = Outer Radius
  • r = Inner Radius

Real-World Applications of Star Shapes

  • Symbolism and Vexillology (Flags)
  • Art and Design
  • Architecture and Decoration
Star polygons are not just abstract geometric figures; they are found in many aspects of human culture and design.
Flags and Symbols
The most common application is in vexillology. Stars on national flags often represent states, unity, or guiding principles. The pentagram and hexagram are powerful symbols in various cultures and belief systems.
Design and Architecture
In graphic design, stars are used to draw attention, rate quality, or add a festive flair. In architecture, star motifs can be found in window designs (rose windows), floor tilings, and decorative elements, especially in historical and religious buildings.

Application Areas

  • National flags (USA, Australia, etc.)
  • Company logos
  • Holiday decorations

Frequently Asked Questions (FAQ)

  • What is the difference between a star polygon and a star-shaped polygon?
  • Why does the star need at least 5 points?
  • What is the 'golden ratio' pentagram?
Can a star have 4 points?
Geometrically, a 4-pointed star is typically just two intersecting lines or a compound of two digons, and doesn't form a continuous boundary like a star polygon. Therefore, this calculator requires 5 or more points to define a valid star polygon.
What is a 'golden ratio' pentagram?
A special type of 5-pointed star (pentagram) is one where the ratio of the outer radius to the length of a segment is the golden ratio (φ ≈ 1.618). This creates a particularly harmonious and aesthetically pleasing shape, found frequently in art and nature. This occurs when the ratio of the inner to outer radius (r/R) is approximately 0.382.