Understanding the Problem
To find the length of OM (where O is the center of a regular polygon), we first need to understand the relationship between the area of the polygon and its dimensions.
Area of Regular Polygon
The area of a regular polygon can be calculated using the formula:
Area = (1/2) × Perimeter × Apothem
For our case, we know the area is 75√3 cm².
Finding the Length of OM
1. Identify Apothem:
The apothem (OM) of the polygon is the perpendicular distance from the center to the midpoint of a side.
2. Using Area Formula:
Rearranging the area formula gives us:
Apothem = (2 × Area) / Perimeter
3. Regular Polygon Properties:
For regular polygons, the perimeter can be expressed as:
Perimeter = number of sides × length of one side
4. Assuming a Triangle:
If we assume the polygon is a regular triangle (for simplification), we can calculate the apothem. The perimeter and area will guide us to find OM.
Calculating OM for a Triangle
1. Area of Triangle:
Area = (√3/4) × side², equating to 75√3 gives:
Side² = 300,
Side = 10√3 cm.
2. Finding Perimeter:
Perimeter = 3 × 10√3 = 30√3 cm.
3. Calculating Apothem:
OM = (2 × 75√3) / (30√3) = 5 cm.
Conclusion
Thus, the length of OM (the apothem) in a regular polygon with an area of 75√3 cm² is 5 cm.