When a regular polygon is inscribed in a circle, the circle is divided into arcs of equal size. The angular size of these arcs is simply related to the size of the interior angles of the polygon. Describe the relationship.
s = rθ = r*2π/n
for an n-gon
The relationship between the angular size of the arcs formed by a regular polygon inscribed in a circle and the size of the interior angles of the polygon can be described as follows:
1. The total angle around a point in a circle is always 360 degrees.
2. For a regular polygon, all interior angles are congruent (equal in size).
3. Let's consider a regular polygon with n sides. Each interior angle of the polygon measures (n-2) * 180 / n degrees.
4. When a regular polygon is inscribed in a circle, the arcs between the vertices of the polygon are formed.
5. The angular size of each arc is equal to the measure of each interior angle of the polygon.
6. Therefore, each arc formed by the regular polygon inscribed in a circle has an angular size of (n-2) * 180 / n degrees.
In summary, the angular size of the arcs formed when a regular polygon is inscribed in a circle is equal to the measure of each interior angle of the polygon, which can be calculated using (n-2) * 180 / n degrees, where n represents the number of sides of the regular polygon.
To understand the relationship between the angular size of the arcs and the interior angles of a regular polygon inscribed in a circle, we need to consider a few key concepts.
1. Central angle: The central angle is the angle formed by two radii of the circle connecting the center of the circle to two adjacent vertices of the polygon. It can be measured in degrees or radians.
2. Interior angle: The interior angle of a regular polygon is the angle formed by two adjacent sides of the polygon. In a regular polygon, all interior angles have equal measures.
Now, let's explain the relationship step by step:
1. Start with a regular polygon inscribed in a circle. Each vertex of the polygon touches the circumference of the circle.
2. Draw radii from the center of the circle to each vertex of the polygon. Since the circle is divided into arcs of equal size, each arc corresponds to a side of the polygon.
3. Notice that each central angle formed by a radius and two adjacent vertices of the polygon is equal. This is because the polygon is regular, meaning all of its sides and angles are congruent.
4. Now, consider a single central angle of the regular polygon. The size of this central angle is equal to the size of the corresponding interior angle of the polygon.
5. Since the circle is divided into arcs of equal size, the measure of each arc is proportional to its corresponding central angle. This means that the angular size of the arcs, measured in degrees or radians, is also equal to the size of the interior angle of the regular polygon.
In summary, the relationship between the angular size of the arcs and the interior angles of a regular polygon inscribed in a circle is that they have equal measures. The angular size of each arc is equal to the size of the corresponding interior angle of the polygon.