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Understanding Convex Polygons: Definition and Properties

CEO Khai Intela
Convex polygons are fascinating closed figures that possess unique characteristics. In this article, we will explore the definition and properties of convex polygons, as well as the differences between convex and concave shapes. What is...

Convex polygons are fascinating closed figures that possess unique characteristics. In this article, we will explore the definition and properties of convex polygons, as well as the differences between convex and concave shapes.

What is a Convex Polygon?

A convex polygon is a two-dimensional shape with straight lines and exterior angles that point outwards. Unlike concave polygons, each interior angle of a convex polygon is less than 180°. This distinctive feature gives convex polygons their name. Take a look at the image below to visualize convex polygons in real-world examples.

Examples of Convex Polygon Examples of Convex Polygon

Properties of Convex Polygons

To easily recognize convex polygons, we can look for the following characteristics:

  • Convex polygons have interior angles that are all less than 180°.
  • Diagonals of a convex polygon lie entirely inside the polygon.
  • Lines joining any two points of a convex polygon are located within the shape itself.

Regular vs. Irregular Convex Polygons

Convex polygons can be further categorized into two types: regular and irregular.

Regular Convex Polygon

A regular convex polygon has sides of equal length, and all of its interior angles are equivalent and less than 180°. Additionally, the vertices of a regular convex polygon are equidistant from its center. For instance, a regular convex pentagon exemplifies this category.

Irregular Convex Polygon

On the other hand, an irregular convex polygon features sides of unequal length and interior angles of unequal measure. There is no symmetry or uniformity in the irregular convex polygon. Examples include an irregular parallelogram.

Understanding the Differences between Convex and Concave Shapes

It is essential to differentiate between convex and concave shapes. Let's explore their distinctive qualities:

Convex Polygon

  • The full outline of a convex shape points outwards, with no dents or inwardly curved sections.
  • All the interior angles of a convex polygon are less than 180°.
  • The line joining any two vertices of a convex shape lies completely within the shape.

Concave Polygon

  • At least some portion of a concave shape points inwards, forming a dent or an inwardly curved area.
  • At least one interior angle of a concave polygon is greater than 180°.
  • The line joining any two vertices of a concave shape may or may not lie within the shape.

Formulas for Convex Polygons

Formulas help us calculate various properties of convex polygons. Let's explore some essential formulas:

Area of a Convex Polygon

The area of a convex polygon is the space covered inside its boundaries. It can be calculated using the following formula:

Area = 1/2 * |(x1y2 - x2y1) + (x2y3 - x3y2) + ... + (xny1 - x1yn)|

Sum of Interior Angles

The sum of the interior angles of a convex polygon with 'n' sides can be determined using the formula:

Sum of Interior Angles = 180 * (n - 2)°

For example, a hexagon has 6 sides, so its sum of interior angles is 180 * (6 - 2)°, which equals 720°.

Sum of Exterior Angles

The sum of the exterior angles of a convex polygon is equal to 360° divided by the number of sides (n) of the polygon.

Conclusion

Convex polygons are captivating geometric shapes with unique properties. By understanding their characteristics and using the appropriate formulas, we can explore their properties and calculate essential measurements. Remember, convex polygons have their interior angles less than 180°, while concave polygons feature at least one interior angle greater than 180°. Explore related topics and expand your knowledge of polygons.

Topics Related to Convex Polygon:

  • Definition of Polygon
  • Polygon Shape
  • Similarity in Triangles
  • Areas of Similar Triangles
  • What is Similarity?
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