To find the measure of one angle of a regular polygon, we can use the formula for the interior angle of a regular polygon, which is given by:
\[ \text{Interior angle} = \frac{(n-2) \times 180°}{n} \]
where \( n \) is the number of sides of the polygon.
For a 40-sided regular polygon, we have \( n = 40 \). Plugging this into the formula, we get:
\[ \text{Interior angle} = \frac{(40-2) \times 180°}{40} \]
This simplifies to:
\[ \text{Interior angle} = \frac{38 \times 180°}{40} \]
Calculating \( 38 \times 180° \):
\[ 38 \times 180° = 6840° \]
Now, we divide by 40:
\[ \text{Interior angle} = \frac{6840°}{40} = 171° \]
Thus, the degree measure of one angle of a 40-sided regular polygon is:
\[ \boxed{171°} \]