Outer Angle Of Polygram Given Chord Length Calculator

Formula Used:

\[ \text{Outer Angle} = \arccos\left(\frac{(2 \times \text{Edge Length}^2) - \text{Chord Length}^2}{2 \times \text{Edge Length}^2}\right) \]

Outer Angle of Polygram:

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1. What is the Outer Angle of Polygram?

The Outer Angle of Polygram is the angle between any two adjacent isosceles triangles which forms the spikes of the Polygram. It is a fundamental geometric property that helps define the shape and symmetry of polygram figures.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ \text{Outer Angle} = \arccos\left(\frac{(2 \times \text{Edge Length}^2) - \text{Chord Length}^2}{2 \times \text{Edge Length}^2}\right) \]

Where:

Edge Length - Length of any edge of the Polygram shape
Chord Length - Distance between any two adjacent spike tips of the Polygram
arccos - Inverse cosine function

Explanation: This formula derives from trigonometric relationships in the isosceles triangles that form the polygram's spikes, using the cosine rule to relate edge lengths to angles.

3. Importance of Outer Angle Calculation

Details: Calculating the outer angle is essential for geometric analysis, architectural design, and understanding the symmetry properties of polygram shapes. It helps in determining the precise angular relationships between different components of the figure.

4. Using the Calculator

Tips: Enter both edge length and chord length in meters. Both values must be positive numbers. The chord length should be less than or equal to twice the edge length for valid results.

5. Frequently Asked Questions (FAQ)

Q1: What is the range of valid inputs for this calculation?
A: Both edge length and chord length must be positive numbers. The chord length cannot exceed twice the edge length for the arccos function to produce a valid result.

Q2: In what units should I input the measurements?
A: The calculator uses meters as the default unit, but you can use any consistent unit system as the formula is dimensionally consistent.

Q3: What types of polygrams does this formula apply to?
A: This formula applies to regular polygrams where all edges and angles are equal, forming symmetrical star-shaped figures.

Q4: Why does the result sometimes show an error message?
A: The error occurs when the input values produce a result outside the domain of the arccos function (-1 to 1), which happens when chord length > 2 × edge length.

Q5: Can this calculator be used for irregular polygrams?
A: No, this formula is specifically designed for regular polygrams where all edges and angles are equal. Irregular polygrams require different calculations.