Inner Angle Of Polygram Given Base Length Calculator

Formula Used:

\[ \text{Inner Angle of Polygram} = \arccos\left(\frac{(2 \times \text{Edge Length of Polygram}^2) - \text{Base Length of Polygram}^2}{2 \times \text{Edge Length of Polygram}^2}\right) \]

Inner Angle of Polygram:

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1. What is the Inner Angle of Polygram?

The Inner Angle of Polygram is the unequal angle of the isosceles triangle which forms the spikes of the Polygram or the angle inside the tip of any spike of Polygram. It is a fundamental geometric property that helps define the shape and symmetry of polygram structures.

2. How Does the Calculator Work?

The calculator uses the mathematical formula:

\[ \angle_{Inner} = \arccos\left(\frac{(2 \times le^2) - lBase^2}{2 \times le^2}\right) \]

Where:

\( \angle_{Inner} \) — Inner Angle of Polygram (in radians)
\( le \) — Edge Length of Polygram (m)
\( lBase \) — Base Length of Polygram (m)

Explanation: This formula derives from the cosine rule applied to the isosceles triangles that form the polygram's spikes, relating the edge lengths to the internal angles.

3. Importance of Inner Angle Calculation

Details: Calculating the inner angle is crucial for geometric design, architectural planning, and understanding the symmetry properties of polygram structures. It helps determine the precise shape characteristics and ensures proper construction of polygram-based designs.

4. Using the Calculator

Tips: Enter both edge length and base length in meters. All values must be positive numbers. The calculator will compute the inner angle in degrees.

5. Frequently Asked Questions (FAQ)

Q1: What is a polygram?
A: A polygram is a star-shaped polygon formed by connecting non-adjacent vertices of a regular polygon, creating intersecting line segments and spikes.

Q2: Why does the formula use arccos function?
A: The arccos function is used because the formula derives from the cosine rule in trigonometry, which relates the sides of a triangle to its angles.

Q3: What are typical values for the inner angle?
A: The inner angle typically ranges between 0° and 180°, depending on the relative lengths of the edges and base of the polygram.

Q4: Can this calculator handle very small or very large values?
A: The calculator can handle a wide range of values, but the input must satisfy the mathematical constraints of the arccos function (-1 ≤ input ≤ 1).

Q5: How accurate are the results?
A: The results are mathematically precise based on the input values, with the output rounded to 6 decimal places for clarity.