Geometric polygons. Shapes and characteristics

05/05/2026

The online geometric polygon simulations on this page will help you learn more about these important elements of mathematics. You will learn about the geometry of polygons and see some of the different shapes they can take.

This Thematic Unit is part of our Mathematics collection

Apothem

The perpendicular distance from the center of a regular polygon to the midpoint of any of its sides.

Diagonal

A line segment connecting two non-consecutive vertices of a polygon.

External Angle

An angle formed by one side of a polygon and the extension of the adjacent side.

Internal Angle

An angle formed by two adjacent sides of a polygon on its interior part.

Perimeter

The total length of a polygon’s boundary, obtained by summing the measures of all its sides.

Polygon

A closed plane geometric figure formed by a finite sequence of consecutive line segments.

Polygon Center

The interior point of a regular polygon that is equidistant from all its vertices.

Polygon Side

Each of the line segments that make up the boundary or contour of the polygon.

Regular Polygon

A geometric figure that is both equilateral (equal sides) and equiangular (equal angles).

Vertex

The joining point where two sides of a polygon meet and where the angle is formed.

What are geometric polygons

In geometry, a polygon is a plane geometric figure composed of a sequence of consecutive straight segments that enclose a region in the plane. The segments are the sides of the polygon called sides, and the points where they intersect are called vertices.

Regular and irregular geometric polygons

Geometric polygons can be regular or irregular. Regular geometric polygons are those in which all sides and angles are equal. Irregular geometric polygons have at least one side or angle that is not equal.

Elements of a geometric polygon

The geometry of polygons is determined by the following geometric elements, present in all polygons:

Sides. These are the segments that form the polygonal line.

Vertices. These are the points where the sides meet.

Angles. They are the regions of the plane that delimit two sides.

Diagonal. The straight line that joins two non-consecutive vertices.

Perimeter. Sum of the length of all the sides.

Area. The space occupied by the polygon in a plane.

In addition, in a regular polygon, there are the following additional elements:

Center. Point from which all angles and sides are at the same distance.

Radius. The segment joining the center of the polygon with any of its vertices.

Apothem. Segment joining the center of the polygon with the center of any of its sides.

Classification of geometric polygons

Geometric polygons can be classified according to different criteria. Two of the most common is to classify them by their number of sides and by their angles.

According to their sides they are classified as follows:

Triangle, 3 sides. Quadrilateral, 4 sides. Pentagon, 5 sides. Hexagon, 6 sides. Heptagon, 7 sides. Octagon, 8 sides. And so on.

According to their angles:

Concave polygons. They have an angle that measures more than 180°.

Convex polygons. All angles measure less than 180°.

The online geometric polygon simulations below are an exceptional tool for discovering the secrets of polygons and advancing your knowledge of geometry.

Apothem

The perpendicular distance from the center of a regular polygon to the midpoint of any of its sides.

Diagonal

A line segment connecting two non-consecutive vertices of a polygon.

External Angle

An angle formed by one side of a polygon and the extension of the adjacent side.

Internal Angle

An angle formed by two adjacent sides of a polygon on its interior part.

Perimeter

The total length of a polygon’s boundary, obtained by summing the measures of all its sides.

Polygon

A closed plane geometric figure formed by a finite sequence of consecutive line segments.

Polygon Center

The interior point of a regular polygon that is equidistant from all its vertices.

Polygon Side

Each of the line segments that make up the boundary or contour of the polygon.

Regular Polygon

A geometric figure that is both equilateral (equal sides) and equiangular (equal angles).

Vertex

The joining point where two sides of a polygon meet and where the angle is formed.

Explore the exciting STEM world with our free, online, simulations and accompanying companion courses! With them you’ll be able to experience and learn hands-on. Take this opportunity to immerse yourself in virtual experiences while advancing your education – awaken your scientific curiosity and discover all that the STEM world has to offer!

Geometric polygon simulations

Quadrilateral and triangle

A quadrilateral is a four-sided polygon. A triangle is a 3-sided polygon. This simulation allows you to visualize different types of triangles and quadrilaterals. Select the type you are interested in and drag the red dots to change its shape.

Types of quadrilaterals

In this simulation you can move the vertices to form different types of quadrilaterals. Note the names they have according to the arrangement of vertices and sides.

External angles

Check with this simulation what happens to the exterior angles of a polygon if we change the number of sides and observe that they always add up to 360º. Can you explain why?

Polygon inscribed and circumscribed to a circumference

Observe with this simulation how the perimeters of an inscribed polygon and a polygon circumscribed to a circle change. What is the best approximation of the value of the circumference? And the average of both?

Polygonal wheel

Can a non-circular wheel roll? This simulation shows you that a non-circular wheel can roll if the ground has a suitable shape. See how the ground changes as the number of sides of the polygon increases.

Giants of science

“If I have seen further, it is by standing on the shoulders of giants”

Isaac Newton

Evangelista Torricelli

1608

–

1647

Evangelista Torricelli invented the mercury barometer, demonstrating the existence of atmospheric pressure, and made advances in hydrodynamics, geometry, and applied optics

“We have only today; let us make good use of this day”

Euclides

300 a.C.

–

Euclid established the foundations of geometry, organizing axioms and theorems

“There is no royal road to geometry”

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Giants of science

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Isaac Newton

Leonhard Euler

1707

–

1783

Euler developed much of modern mathematical notation and made key contributions to analysis, topology, mechanics, and graph theory

“Nothing takes place in the universe without mathematics expressing it”

Pythagoras

570 a.C.

–

495 a.C.

Pythagoras established the famous theorem on right triangles and founded a school linking mathematics, music, and astronomy as fundamental principles of the universe

“All is number”

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Test your knowledge

What is a polygon and what are its main elements?

A polygon is a closed flat figure formed by consecutive straight segments called sides. The main elements of a polygon are the sides, which form the segments that enclose the figure; the vertices, which are the points where two sides meet; and the interior angles, formed at each vertex between adjacent sides. Additionally, polygons can be classified according to the number of sides—triangle, quadrilateral, pentagon, etc.—and by other properties such as side lengths and angle measures. Understanding these elements is key for analyzing and solving geometric problems involving measurements, symmetry, and spatial relationships.

How are polygons classified and what are the differences between them?

Polygons can be classified mainly by the number of sides and by their regularity. According to the number of sides, they are named triangle (3), quadrilateral (4), pentagon (5), hexagon (6), and so on. A polygon is considered regular if all its sides and all its interior angles are equal; if not, it is irregular. This classification helps distinguish figures with very different shapes and properties, and it is fundamental for understanding how they behave in calculations of perimeter, area, and symmetry.

How is it that a triangle, with only three sides, can be so useful for building other shapes?

It makes a lot of sense that triangles are so special because, being the simplest figure that encloses an area, they serve as a base for constructing more complex shapes. In many practical applications—like structures and design—triangles provide stability because, being rigid, they don’t change shape unless their sides are deformed, unlike figures with more sides, which can be more flexible.

What happens to the angles when you increase the number of sides of a polygon?

When you increase the number of sides of a polygon, the sum of the interior angles also increases. For example, a triangle has a sum of 180°, a quadrilateral 360°, and a pentagon even more. This happens because every new side adds a vertex that contributes an interior angle, and this sum is reflected in the overall geometry of the figure.

Does it make sense that a regular polygon is different from an irregular one even if they have the same number of sides?

Yes, it makes sense because, even though both have the same number of sides, the uniformity of measurements completely changes their properties. In a regular polygon, all sides and angles are equal, giving the figure symmetry and making it easier to analyze. In an irregular polygon, the lack of uniformity can result in very different sides and angles, affecting aspects like area calculation or practical applications of the shape.