How do I find the radius of a regular nonagon with an apothem of length 6.42in.
There are 9 40° angles in a circle, so the nonagon consists of 9 isosceles triangles, having angles of 40,70,70.
The radius is r, where
6.42/r = cos20°
To find the radius of a regular nonagon (a nine-sided polygon) with an apothem of length 6.42 inches, you can use the formula for the radius of a regular polygon:
Radius = Apothem / Cos (180° / Number of sides)
In this case, the number of sides is 9 because it is a regular nonagon. So the formula becomes:
Radius = 6.42 / Cos(180° / 9)
Now let's calculate it step by step:
1. Convert the angle from degrees to radians:
180° / 9 = 20°
To convert degrees to radians, you need to multiply by π/180. So:
20° * π/180 ≈ 0.3491 radians
2. Calculate the cosine of 0.3491 radians:
Cos(0.3491) ≈ 0.9397
3. Divide the apothem length by the cosine value:
Radius = 6.42 / 0.9397
Now, perform the division:
Radius ≈ 6.84 inches
Therefore, the radius of the regular nonagon with an apothem of length 6.42 inches is approximately 6.84 inches.