The maximum number of right angles a hexagon can possess is five.
Understanding Hexagon Geometry
A hexagon is a polygon characterized by its six sides and six interior angles. A fundamental geometric principle states that the sum of the interior angles of any polygon with n sides is given by the formula (n-2) 180°. For a hexagon, where n=6, the sum of its interior angles is (6-2) 180° = 4 * 180° = 720 degrees. This fixed sum is crucial for determining the possible number of right angles.
Why Five is the Absolute Maximum
Let's analyze the possibilities based on the total sum of interior angles (720°):
-
Six Right Angles (6 x 90°):
- If a hexagon were to have six right angles, the sum of its interior angles would be 6 * 90° = 540°.
- This sum (540°) is less than the required 720° for any valid hexagon. It's geometrically impossible for a polygon with six sides to have all its interior angles measuring 90 degrees, as the sides would not connect to form a closed shape that satisfies the 720° angle sum. Therefore, a hexagon cannot have six right angles.
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Five Right Angles (5 x 90°):
- If a hexagon has five right angles, their combined sum is 5 * 90° = 450°.
- To meet the total sum of 720°, the sixth and final angle must be 720° - 450° = 270°.
- A 270-degree angle is a valid interior angle for a polygon. It is classified as a reflex angle (an angle greater than 180° but less than 360°), which means the hexagon would be non-convex (also known as concave). This configuration allows for the polygon to close properly while respecting all geometric rules.
This analysis clearly demonstrates that five right angles is the highest number achievable for a hexagon.
Example Angle Distribution
A hexagon with five right angles would typically have an angle distribution similar to this:
| Angle Number | Measurement | Type of Angle |
|---|---|---|
| Angle 1 | 90° | Right Angle |
| Angle 2 | 90° | Right Angle |
| Angle 3 | 90° | Right Angle |
| Angle 4 | 90° | Right Angle |
| Angle 5 | 90° | Right Angle |
| Angle 6 | 270° | Reflex Angle |
| Total Sum | 720° |
This combination perfectly satisfies the geometric requirements for a hexagon.
Constructing a Five Right-Angle Hexagon
Such a hexagon is a common example of a non-convex polygon. You can visualize it by imagining a large rectangle, and then "cutting out" a smaller rectangular section from one of its sides. For instance, consider a rectangle and imagine a square-shaped bite taken out of its edge. The resulting shape would have an interior angle that "turns inwards," forming the 270° reflex angle, while the other five angles around the perimeter remain 90°.
A simple coordinate example of such a hexagon:
- (0,0)
- (5,0) (90° at (0,0))
- (5,3) (90° at (5,0))
- (2,3) (90° at (5,3))
- (2,5) (90° at (2,3))
- (0,5) (90° at (2,5))
- The final angle at (0,5), connecting back to (0,0) and to (2,5), would be the 270° reflex angle.
Key Considerations for Polygon Angles
- Convex vs. Non-convex: A hexagon with five right angles must be non-convex (concave) because it requires at least one interior angle greater than 180° (the 270° reflex angle). A convex polygon has all interior angles less than 180°.
- Angle Limits: While individual interior angles can vary, they must always be less than 360°. A 360° angle implies a full circle, which is not a valid interior angle for a simple polygon.
Understanding these properties is crucial for various applications, from architectural design to geometric modeling, where specific polygonal shapes with defined angles are often required.
