Lesson 19: Polygons

New Concept

This course builds your mathematical foundation step-by-step. We begin with fundamental shapes like polygons, which are closed plane figures with straight sides.

What’s next

This lesson is your starting point. Next, you'll master how to name, classify, and compare different polygons using concepts like congruence and similarity.

Property

Polygons are closed plane figures with straight sides. They have the same number of sides as angles. A polygon is regular if all its sides are the same length and all its angles are the same size.

Examples

A triangle is a polygon, but a circle is not because it has a curved side.
A stop sign is a regular octagon because it has 8 equal sides and 8 equal angles.
A rectangle is a polygon, but it's only regular if it's also a square.

Explanation

Think of a polygon as a completely fenced-in yard made only with straight fence pieces. If all the fence pieces are the same length and all the corners have the same angle, you’ve got a fancy 'regular' polygon! No curves or open gates are allowed in this exclusive geometry club.

Property

Figures are congruent (≅) if they are the same shape and size. Figures are similar (~) if they have the same shape but can be different sizes. One figure can be a dilation (enlargement) of another.

Examples

Two identical squares are congruent: $\text{Square A} \cong \text{Square B}$.
A small triangle and a larger version of it are similar: $\text{Triangle A} \sim \text{Triangle C}$.
All circles are similar to each other, but they are only congruent if their radii are equal.

Explanation

Congruent figures are like identical twins—perfect matches. Similar figures are like a photo and its enlargement; they look alike with the same angles but come in different sizes. One is often just a scaled-up or scaled-down version of the other. It's all in the family, but they're not identical!

Property

The point where two sides of a polygon meet is called a vertex (plural: vertices). A particular polygon can be identified by naming the letters of its vertices in order.

Examples

In a square named $ABCD$, the vertices are the points $A$, $B$, $C$, and $D$.
A pentagon has 5 sides and therefore also has 5 vertices.
You can name a triangle with vertices $X, Y, Z$ as $\triangle XYZ$ or $\triangle ZYX$.

Explanation

A vertex is just a fancy word for a corner! It's the spot where two straight sides meet up to say hello. To name a polygon, you just play connect-the-dots with the vertices' letter-names, either clockwise or counter-clockwise. It's like calling out the stops on a geometric train ride!