To find the angles of rotation that would carry a regular octagon onto itself, we can use the formula for the angle of rotation, which is given by:
\[ \text{Angle of rotation} = \frac{360°}{n} \]
where \(n\) is the number of sides of the polygon. For a regular octagon, \(n = 8\).
Calculating that gives:
\[ \text{Angle of rotation} = \frac{360°}{8} = 45° \]
This means that rotating the octagon by 45 degrees will map it onto itself.
Next, we need to check the multiples of 45 degrees that can also map the octagon onto itself:
- \(45° \times 1 = 45°\)
- \(45° \times 2 = 90°\)
- \(45° \times 3 = 135°\)
- \(45° \times 4 = 180°\)
- \(45° \times 5 = 225°\)
- \(45° \times 6 = 270°\)
- \(45° \times 7 = 315°\)
- \(45° \times 8 = 360°\) (which is a full rotation)
From these, the valid rotation angles within a full 360-degree rotation that will map the octagon onto itself are \(45°\), \(90°\), \(135°\), \(180°\), \(225°\), \(270°\), and \(315°\).
Among the given choices, 270° is one of the angles of rotation that will carry a regular octagon onto itself.
Thus, the correct answer is:
270°
2 answers