To find the measure of one interior angle of a regular 23-gon, we can use the formula for calculating the measure of an interior angle of a regular polygon. Here is the step-by-step explanation:
1. Understand the formula:
The formula to find the measure of one interior angle of a regular polygon with [tex]\( n \)[/tex] sides is:
[tex]\[
\text{Interior angle} = \frac{(n-2) \times 180^\circ}{n}
\][/tex]
This formula comes from the fact that the sum of all interior angles of a polygon with [tex]\( n \)[/tex] sides is [tex]\((n-2) \times 180^\circ\)[/tex].
2. Substitute the given value:
For a 23-gon, [tex]\( n = 23 \)[/tex]. Substitute this value into the formula:
[tex]\[
\text{Interior angle} = \frac{(23-2) \times 180^\circ}{23}
\][/tex]
3. Simplify the expression:
Calculate the numerator first:
[tex]\[
23 - 2 = 21
\][/tex]
So, the numerator becomes:
[tex]\[
21 \times 180^\circ
\][/tex]
Now, calculate:
[tex]\[
21 \times 180^\circ = 3780^\circ
\][/tex]
4. Divide by the number of sides:
Next, divide this result by the number of sides, which is 23:
[tex]\[
\frac{3780^\circ}{23}
\][/tex]
Perform the division:
[tex]\[
3780^\circ \div 23 = 164.34782608695653^\circ
\][/tex]
Therefore, the measure of one interior angle of a regular 23-gon is approximately [tex]\( 164.3^\circ \)[/tex].
So, the correct answer is:
D. 164.3
