How to Inscribe a Regular Polygon within a Circle

• A straightedge or ruler for accurate linear measurements.
• A compass for drawing the circle and aiding in polygon construction.
• A pencil for drawing and annotations.

Examples

Practice Questions:

1. What would be the central angle for a regular decagon (\(10\) sides)?
2. How many vertices will touch the circle if a regular pentagon is inscribed in it?
3. For a given central angle, can you determine the number of sides of the inscribed regular polygon?

1. For a decagon, the central angle is \( \frac{360^\circ}{10} = 36^\circ \).
2. A regular pentagon has \(5\) vertices, so all 5 vertices will touch the circle.
3. Yes, using the formula \( n = \frac{360^\circ}{\text{Central Angle}} \). The value of \( n \) will give the number of sides of the polygon.

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