3.1.1. Convex and concave polygons

A simple closed curve made up of only line segments is called a polygon.

A polygon is a geometric shape formed by connecting a sequence of points, known as vertices, with straight line segments. The line segments are called sides, and the vertices are the corners of the polygon. When the shape is closed, meaning all the sides connect back to the starting point without any gaps, it's defined as a simple closed curve. Polygons can have different numbers of sides, such as triangles (3 sides), quadrilaterals (4 sides), and so on.

Imagine a shape you might see drawn on a paper, like a drawing of a house. The outlines of the house that connect the corners are polygons. The four walls together with the roof create a closed shape and represent a polygon.

Polygons that are convex have no portions of their diagonals in their exteriors or any line segment joining any two different points in the interior of the polygon lies wholly in the interior of it. This is not true with concave polygons.

A convex polygon is one where, when you draw a line between any two points within the shape, the line remains inside the polygon. No part of the shape will 'cave in' towards the inside. In contrast, a concave polygon has at least one internal angle greater than 180 degrees, meaning that when you connect two points within the shape, the line may pass outside the polygon's edges.

Think about a tent. If all the edges of the tent are stretched outward, it forms a convex shape. However, if part of the tent is pushed inward, creating a 'cave,' it represents a concave polygon. The convex shape keeps everything inside while the concave one has parts where the edges bend inward.

Study the figures given. Then try to describe in your own words what we mean by a convex polygon and what we mean by a concave polygon. Give two rough sketches of each kind.

To understand the differences between convex and concave polygons visually, look at illustrations or create sketches yourself. A convex polygon might look like a regular hexagon where every angle is less than 180 degrees. Meanwhile, a concave polygon could be a star shape or any figure that has at least one inward dent. Describing these shapes helps solidify your understanding of polygons in two dimensions.

Imagine the shapes of various objects. A bookshelf is a good example of a convex shape: its front is entirely outward. Now, think about a wedge of cheese, where one side is pushed in—that creates a concave shape. Visualizing everyday objects can deepen your grasp of these concepts.

In our work in this class, we will be dealing with convex polygons only.

In this lesson, the focus will be solely on convex polygons because they are generally simpler to work with for calculations involving angles, perimeter, and area. Convex polygons do not present complications like concave polygons, where lines might crisscross outside the shape, making calculations more complex.

When building models or structures, architects often prefer convex shapes because they offer better stability and are easier to manage. For example, when designing parks, which often have paths that are simple to follow and enclosed spaces, convex shapes like circles or ovals are favored.

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