⭐ Star Shape Calculator
Calculate properties of regular star polygons
Common: 5 (pentagram), 6, 7, 8 points
Distance from center to outer points
Distance from center to inner points
How to Use This Calculator
Enter Number of Points
Input the number of points (arms) of the star. Common values: 5 (pentagram), 6, 7, or 8 points.
Enter Radii
Input the outer radius (distance from center to outer points) and inner radius (distance from center to inner points). Outer radius must be greater than inner radius.
Click Calculate
Press the "Calculate Star Properties" button to get the area and perimeter of the star shape.
Formula
Area = (n/2) × R × r × sin(π/n)
Perimeter = n × [2R sin(π/n) + 2r sin(π/n)]
Where:
- n = Number of points (arms)
- R = Outer radius (distance to outer points)
- r = Inner radius (distance to inner points)
- π ≈ 3.14159
Example (5-pointed star): R = 10, r = 4
Area = (5/2) × 10 × 4 × sin(π/5)
Area = 100 × sin(36°) = 100 × 0.588 = 58.8 square units
Perimeter = 5 × [2×10×sin(36°) + 2×4×sin(36°)] = 5 × [20×0.588 + 8×0.588] = 82.32 units
About Star Shape Calculator
The Star Shape Calculator helps you find the area and perimeter of regular star polygons. A star polygon is formed by connecting vertices of a regular polygon in a specific pattern, creating a star-like shape.
When to Use This Calculator
- Design: Calculate star dimensions for logos, graphics, and decorative elements
- Architecture: Design star-shaped structures, windows, or architectural features
- Art & Crafts: Plan star-shaped projects and calculate material requirements
- Education: Learn and practice star polygon mathematics
- Engineering: Design star-shaped components and mechanical parts
- Flag Design: Calculate star dimensions for flag designs and patterns
Why Use Our Calculator?
- ✅ Flexible Points: Supports any number of star points (3+)
- ✅ Complete Properties: Calculates area and perimeter
- ✅ Instant Results: Get all measurements immediately
- ✅ Step-by-Step Display: See calculation formulas
- ✅ Works with Any Units: Meters, feet, inches, or any unit
- ✅ 100% Accurate: Precise geometric calculations
- ✅ Completely Free: No registration required
Understanding Star Shapes
Star polygons are formed by connecting vertices of regular polygons. Key concepts:
- Points: Number of arms or points of the star
- Outer Radius: Distance from center to the outer (pointed) vertices
- Inner Radius: Distance from center to the inner (concave) vertices
- Pentagram: 5-pointed star (most common, used in flags and symbols)
- Symmetry: Star polygons have rotational symmetry based on number of points
- Golden Ratio: 5-pointed stars (pentagrams) often relate to the golden ratio
Real-World Applications
Flag Design: Many flags feature stars. The US flag uses 50 five-pointed stars. Calculate star dimensions for flag designs and ensure proper proportions.
Logo Design: Star shapes are popular in logos and branding. Calculate area and dimensions for balanced star designs.
Architecture: Star-shaped windows, floor patterns, and decorative elements use star polygon geometry. Calculate dimensions for construction and design.
Frequently Asked Questions
What is a star polygon?
A star polygon is a type of non-convex polygon formed by connecting vertices of a regular polygon. It creates a star-like shape with points (arms) extending outward and indentations (concave points) between them.
What is the most common star shape?
The 5-pointed star (pentagram) is most common, appearing in flags, symbols, and designs worldwide. It's formed from a regular pentagon by connecting alternate vertices.
How do I measure the radii of a star?
Outer radius (R): Measure from the center to the tip of any point. Inner radius (r): Measure from the center to the bottom of the indent between two points. Both should be measured along the same radial line.
Can I create stars with any number of points?
Yes, you can create star polygons with 3 or more points. Common values are 5, 6, 7, and 8 points. Very high numbers create more circular, less star-like shapes.
What's the relationship between star and regular polygon?
A star polygon is created from a regular polygon by connecting vertices in a pattern. For example, a pentagram (5-pointed star) is created from a regular pentagon by connecting every other vertex.
Why are 5-pointed stars special?
5-pointed stars (pentagrams) are special because they relate to the golden ratio. The ratio of outer to inner radius in a regular pentagram equals the golden ratio φ ≈ 1.618, making them mathematically and aesthetically significant.
Can the inner radius be zero?
No, the inner radius must be positive and less than the outer radius. If inner radius equals outer radius, you get a regular polygon, not a star. If inner radius approaches zero, the star becomes very "spiky."