How to Inscribe a Regular Polygon within a Circle

To inscribe a regular polygon in a circle, divide the full 360 degrees by the number of sides and mark that angle repeatedly around the center. Connecting the marks gives equal sides and equal angles.

• 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.

Worked examples: inscribing a regular polygon in a circle

Practice questions on inscribed regular polygons

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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