Inner Angle of Polygram given Outer Angle Calculator

Inner Angle of Polygram Formula:

\[ \angle_{Inner} = \angle_{Outer} - \frac{2\pi}{N_{Spikes}} \]

Inner Angle of Polygram:

Unit Converter ▲

Unit Converter ▼

From:	To:

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 an important geometric property that helps define the shape and symmetry of polygrams.

2. How Does the Calculator Work?

The calculator uses the formula:

\[ \angle_{Inner} = \angle_{Outer} - \frac{2\pi}{N_{Spikes}} \]

Where:

\( \angle_{Inner} \) — Inner Angle of Polygram (radians)
\( \angle_{Outer} \) — Outer Angle of Polygram (radians)
\( N_{Spikes} \) — Number of Spikes in Polygram
\( \pi \) — Archimedes' constant (approximately 3.14159)

Explanation: This formula calculates the inner angle by subtracting the angular spacing between spikes from the outer angle of the polygram.

3. Importance of Inner Angle Calculation

Details: Calculating the inner angle is crucial for understanding the geometric properties of polygrams, designing symmetrical patterns, and solving geometric problems involving star-shaped polygons and other complex geometric figures.

4. Using the Calculator

Tips: Enter the outer angle in radians and the number of spikes (must be at least 3). The calculator will compute the corresponding inner angle of the polygram.

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 spikes or star points.

Q2: How is the outer angle defined?
A: The outer angle of polygram is the angle between any two adjacent isosceles triangles which forms the spikes of the Polygram.

Q3: Can this formula be used for any polygram?
A: Yes, this formula applies to all regular polygrams where the spikes are formed by isosceles triangles attached to a regular polygon.

Q4: What are typical values for inner angles?
A: Inner angles typically range from acute to obtuse angles depending on the number of spikes and the outer angle configuration.

Q5: How does the number of spikes affect the inner angle?
A: As the number of spikes increases, the term (2π/N_Spikes) decreases, resulting in a larger inner angle for a given outer angle.