Formula For Finding Diagonals Of A Polygon

Ever sat in a geometry class staring at a shape, trying to figure out how many lines you could draw from one corner to another without just counting them by hand? In practice, it feels like a tedious chore. You draw one, you draw another, you realize you've missed one, and suddenly you're staring at a messy scribble of lines instead of a math problem.

But there is a way to skip the drawing entirely. There is a logic to how these lines connect, and once you see the pattern, you won't need to sketch a single thing.

What Is a Diagonal in a Polygon?

When we talk about a polygon, we're talking about any flat, closed shape made of straight lines—think triangles, squares, pentagons, or even a complex decagon. Practically speaking, a diagonal isn't just any line. It’s a specific type of connection.

The Connection Rule

To be a diagonal, a line segment must connect two corners (vertices) that are not already connected by an edge. If you draw a line from one corner to the very next corner, you haven't made a diagonal; you've just traced the perimeter of the shape. A diagonal has to "cut through" the interior of the polygon.

Why This Matters

Understanding diagonals isn't just about passing a test. It’s about understanding the internal structure of shapes. In computer graphics, for example, breaking complex polygons down into triangles (which have zero diagonals) is a fundamental way software renders 3D objects on your screen. In architecture, the number of diagonals can dictate how much structural bracing a shape might need to remain rigid.

Why the Formula Exists

If you want to find the number of diagonals, you don't actually need to memorize a random string of letters and numbers. You just need to understand how vertices behave.

Think about it this way: every corner in a polygon has the potential to connect to every other corner. If you have a shape with $n$ sides, it also has $n$ corners.

The Logic of Connections

If you stand at one corner of a pentagon, you can theoretically draw a line to four other corners. But, as we mentioned earlier, you can't draw a diagonal to yourself, and you can't draw a diagonal to the two corners immediately next to you. That leaves you with only two possible diagonals from that single point.

This logic applies to every corner. But there's a catch. That said, if you multiply the number of corners by the number of available connections from each corner, you get a total. If you draw a line from Corner A to Corner C, and then later you draw a line from Corner C to Corner A, you've counted the exact same line twice.

That's why we have to divide by two. We're correcting for that double-counting.

The Formula for Finding Diagonals of a Polygon

Here is the math that makes the drawing unnecessary. To find the number of diagonals ($d$) in a polygon with $n$ sides, use this:

$d = \frac{n(n - 3)}{2}$

Let's break down why that $n - 3$ is there, because that's where most people trip up.

Breaking Down the Variables

The $n$ represents the total number of vertices (corners) in your polygon. The $(n - 3)$ is the "available connections" part. Why 3? Because from any given corner, you cannot draw a diagonal to:

Itself.
The neighbor to the left.
The neighbor to the right.

That’s three points off the table for every single vertex.

Real-World Examples

Let's test it to make sure it actually works.

The Square (Quadrilateral) A square has 4 sides. $4 \times (4 - 3) = 4 \times 1 = 4$. Divide that by 2, and you get 2. If you visualize a square, you can clearly see only two diagonals crossing in the middle. It works.

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The Pentagon A pentagon has 5 sides. $5 \times (5 - 3) = 5 \times 2 = 10$. Divide by 2, and you get 5. A pentagon has 5 diagonals, forming that star shape inside.

The Decagon What if we have a 10-sided shape? $10 \times (10 - 3) = 10 \times 7 = 70$. Divide by 2, and you get 35. Trying to draw 35 lines inside a shape by hand would be a nightmare, but the formula handles it in seconds.

Common Mistakes / What Most People Get Wrong

Even with a formula, it's easy to stumble. I've seen students—and even professionals—make these specific errors.

Forgetting the Division

The most common mistake is calculating $n(n - 3)$ and stopping there. If you do this, you are counting every diagonal twice. You'll end up with double the actual amount. Always remember that a diagonal between Point A and Point B is the same as the one between Point B and Point A.

The Triangle Trap

People often try to apply the formula to a triangle and get confused when the answer is zero. $3 \times (3 - 3) = 3 \times 0 = 0$. The math actually works perfectly, but it's a good mental check: a triangle has no diagonals because every corner is already connected to every other corner by an edge. There are no "non-adjacent" corners to connect to.

Confusing Sides with Vertices

In a simple polygon, the number of sides and the number of vertices are the same. But when you get into more complex, non-convex, or "self-intersecting" polygons, things get weird. For standard, convex polygons (the ones that look "normal"), $n$ is both sides and vertices. Just keep that distinction in mind if you move into higher-level geometry.

Practical Tips / What Actually Works

If you are working through a series of geometry problems, don't just rely on the formula blindly. Use these strategies to stay accurate.

Use Visual Verification for Small Shapes

If you are dealing with a quadrilateral, pentagon, or hexagon, do a quick sketch. It takes five seconds. If your formula says 5 and your sketch shows 5, you're golden. If your sketch shows 2, you know you forgot to divide by 2. This is a "sanity check" that prevents silly errors.

The "n-3" Mental Shortcut

Instead of writing out the whole formula every time, just remember: "Multiply the sides by the sides minus three, then cut it in half." It’s a much faster way to process the information in your head.

Dealing with Complex Polygons

If you encounter a shape that isn't "convex"—meaning it has a dent in it (like a star shape)—the formula for the total* number of diagonals still works, but some of those diagonals will actually fall outside* the shape. If a question asks for "internal diagonals," you have to be careful. For standard school-level geometry, you're usually dealing with convex polygons, so you don't need to worry about this unless you're getting into advanced topology.

FAQ

Can a polygon have a negative number of diagonals?

No. Since a polygon must have at least 3 sides, the value of $(n - 3)$ will always be 0 or greater. If you get a negative number, you've likely made a calculation error or used the wrong value for $n$.

Does the formula work for regular polygons?

Yes. A regular polygon (where all sides and angles are equal, like a square or a regular pentagon) follows the same rules as irregular polygons. The formula cares about the number of vertices, not the length of the sides or the measure of the angles.

What is the difference between a diagonal and a chord?

This is a common point of confusion. A diagonal is a term used for polygons (shapes made of straight lines). A chord is a term used for circles (the line segment connecting two points on a circle's circumference). While they are conceptually similar, they belong to different geometric families.