Polygons are an important part of geometry that we encounter every day. They are 2D shapes with straight sides, and their name comes from the Greek words “poly” meaning “many” and “gonia” meaning “angles.”
What is a Polygon?
A polygon is a flat, two-dimensional shape made up of straight line segments that are connected to form a closed figure. Each point where two sides meet is called a vertex, and the line segments are called edges or sides. A polygon must have at least three sides, and all sides must be straight.
Types of Polygons
Polygons are classified based on the number of sides they have. Here are some common types:
Triangle – A polygon with 3 sides.
Quadrilateral – A polygon with 4 sides (like squares and rectangles).
Pentagon – A polygon with 5 sides.
Hexagon – A polygon with 6 sides.
Heptagon – A polygon with 7 sides.
Octagon – A polygon with 8 sides.
Nonagon – A polygon with 9 sides.
Decagon – A polygon with 10 sides.
These are just a few examples—there are many more polygons!
Properties of Polygons
Sides and Angles
A polygon has the same number of sides and angles. For example, a triangle has 3 sides and 3 angles, while a pentagon has 5 sides and 5 angles.
Regular and Irregular Polygons
- A regular polygon has all sides and angles equal, like an equilateral triangle or a square.
- An irregular polygon has sides and angles that are not equal, like a rectangle or a scalene triangle.
Convex and Concave Polygons
- A convex polygon has all its interior angles less than 180°, and no sides “cave in.”
- A concave polygon has at least one interior angle greater than 180°, which gives it a “caved in” appearance.
Sum of Interior Angles
The formula to find the sum of the interior angles of a polygon is
Sum of Interior Angles = (n-2) x 180°
where n is the number of sides.
Angle Between Adjacent Sides
In regular polygons, each angle is equal. The interior angles between adjacent sides in a regular polygon can be calculated using the formula:
Interior Angle = $\frac{(n-2)}{n}$ x 180°
where n is the number of sides.
Diagonals
A diagonal is a line segment connecting two non-adjacent vertices of a polygon. The number of diagonals in a polygon can be calculated using the formula:
Number of Diagonals = $\frac{n(n-3)}{2}$
where n is the number of sides.
The below table lists the common polygons with their number of diagonals, the sum of their interior angles and the angles between the adjacent sides.
Symmetry
Some polygons have lines of symmetry, which means they can be divided into two identical halves. Regular polygons, like squares and equilateral triangles, have the highest degree of symmetry. The number of lines of symmetry in a regular polygon is equal to the number of its sides. For example, a regular pentagon has 5 lines of symmetry, while a regular hexagon has 6.
Real-life Examples
Polygons are all around us! A few examples of polygons are:
- A triangle in a road sign.
- A hexagon in a honeycomb.
- A pentagon in a soccer ball’s design.
- Quadrilaterals like squares and rectangles in tiles and windows.
- An octagon in a stop sign.
Polygons are a key concept in geometry, and understanding them helps us describe the world around us. Whether you’re looking at a wooden door or a tiled floor, you’re seeing polygons in action.
Practice Quiz on Polygons
FAQs on Polygons
- A diagonal is a line segment that connects two non-adjacent vertices of a polygon. For example, in a quadrilateral, the diagonals connect opposite corners.
- A polygon is a flat, 2D shape with straight sides, while a polyhedron is a 3D shape made up of flat polygonal faces, edges, and vertices.
- A self-intersecting polygon is a polygon that crosses over itself. An example is a star shape, which has sides that intersect at points.
- Regular polygons have the following properties:
- All sides are equal in length.
- All interior angles are equal.
- They exhibit symmetry.
- A cyclic polygon is a polygon whose vertices all lie on a single circle. For example, a cyclic quadrilateral has all its vertices on a circle.
- No, polygons cannot have curved sides. All sides must be straight lines. If a shape has curved edges, it is not classified as a polygon.
- All regular polygons are polygons, but not all polygons are regular. Regular polygons have equal sides and angles, while irregular polygons do not.
- Yes, polygons can be classified as symmetric (having at least one line of symmetry) or asymmetric (no lines of symmetry). Regular polygons are typically symmetric.
- A polygon with n sides is referred to as an n-gon. For example, a polygon with 10 sides is called a decagon, and a polygon with 20 sides is called a 20-gon.
- To find the area of irregular polygons, you can divide the shape into smaller regular shapes (triangles, rectangles, etc.), calculate the area of each, and then sum them up.
- The perimeter of a polygon is the total length of all its sides. For a regular polygon, it is calculated as Perimeter=n×s, where n is the number of sides and s is the length of one side.