Question

Draw the following regular polygons inscribed in a circle:

Answer 1

Answer:

• first attachment has pentagon and decagon
• second attachment has hexagon and dodecagon
• computation info explained below

Step-by-step explanation:

1, 2. Central Angle, Interior Angle

See the 3rd attachment for the values. (Angles in degrees.)

The central angle is 360°/n, where n is the number of vertices. For example, the central angle in a pentagon is 360°/5 = 72°.

The interior angle is the supplement of the central angle. For a pentagon, that is 180° -72° = 108°.

These formulas were implemented in the spreadsheet shown in the third attachment.

3. Angles vs. Number of Sides

The size of the central angle is inversely proportional to the number of sides. In degrees, the constant of proportionality is 360°.

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Comment on the drawings

The drawings are made by a computer algebra program that is capable of computing the vertex locations around a unit circle based on the number of vertices. The only "work" required was to specify the number of vertices the polygon was to have. The rest was automatic.

The above calculations describe how the angles are computed. Converting those to Cartesian coordinates for the graphics plotter involves additional computation and trigonometry that are beyond the required scope of this answer.

These figures can be "constructed" using a compass and straightedge. No knowledge of angle measures is required for following the recipes to do that.