Final Answer:
The internal angle of a regular polygon with 40 sides is approximately 171 degrees or 2.98 radians. Since none of the given options match these values, it would seem that the correct option is not listed.
Step-by-step explanation:
To find the internal angle of a regular polygon with 40 sides, we can follow a step-by-step mathematical approach.
First, we calculate the sum of all the internal angles of the polygon using the formula:
Sum of internal angles = (n - 2) * 180 degrees,
where n is the number of sides of the polygon.
For a polygon with 40 sides, we have:
Sum of internal angles = (40 - 2) * 180 degrees
Sum of internal angles = 38 * 180 degrees
Sum of internal angles = 6840 degrees
Now that we have the sum of all internal angles, to find the measure of each individual internal angle, we divide the sum by the number of sides (n), since a regular polygon has sides of equal length and angles of equal measure.
So, the measure of each internal angle is:
Internal angle = Sum of internal angles / n
Internal angle = 6840 degrees / 40
Internal angle = 171 degrees
Now that we have the internal angle in degrees, we can convert this to radians. We know that π radians equal 180 degrees, so to convert degrees to radians, we multiply by π and divide by 180.
Therefore, the internal angle in radians is:
Internal angle in radians = Internal angle in degrees * (π / 180)
Internal angle in radians = 171 * (π / 180)
We substitute π with a numerical approximation, 3.14159, to find the numerical value:
Internal angle in radians = 171 * (3.14159 / 180)
Internal angle in radians ≈ 2.984513 (approximately)
Now we have the values to compare to the multiple-choice options. The closest match to our calculated internal angle of 171 degrees and approximately 2.98 radians is not available in the choices you provided (a through d). It seems there might be a mistake, as none of these options corresponds to the internal angle of a regular polygon with 40 sides.