What is the relationship between the measure of a central
angle of a polygon and the measures of an interior
and an exterior angle of the polygon?
It is important to specify that the following calculations apply to a regular polygon, of which all sides are equal in length, and all interior angles are equal.
Denote the number of sides by n.
The total of the central angles make up 360°, so each central angle is 360/n°.
The exterior angle equals also 360/n°. Note that the exterior angle is the supplement of the interior angle, namely 180° - interior angle.
The interior angle is therefore 180-exterior angle, and in terms of n, it is
interior angle = 180 - 360/n
To understand the relationship between the measure of a central angle of a polygon and the measures of an interior and an exterior angle of the polygon, we need to first define these terms.
- Central angle: A central angle is an angle whose vertex is at the center of a polygon and whose sides extend to the vertices of the polygon. It is measured in degrees.
- Interior angle: An interior angle is an angle formed by two sides of a polygon at a vertex on the inside of the shape. It is also measured in degrees.
- Exterior angle: An exterior angle is an angle formed by one side of a polygon and the extension of an adjacent side. It is measured in degrees.
Now, let's consider a regular polygon, which is a polygon with all sides and angles equal.
In a regular polygon, the central angle, interior angle, and exterior angle have a relationship given by the following formulas:
1. The measure of a central angle is equal to the measure of an exterior angle.
Central Angle = Exterior Angle
2. The measure of an interior angle is equal to the sum of the measures of the central angle and the exterior angle.
Interior Angle = Central Angle + Exterior Angle
3. The measure of an interior angle is also given by the formula:
Interior Angle = (180 * (n - 2)) / n
where n is the number of sides of the polygon.
With these formulas, we can find the relationship between the measures of a central angle, interior angle, and exterior angle of a polygon.