How Do You Find The Exterior Angle Of A Polygon

What Is an Exterior Angle of a Polygon?

Picture a five-sided shape — a pentagon — sitting on a table. At each corner, you turn. Also, that's the exterior angle. You walk around it, tracing the edges. It's one of the most intuitive ideas in geometry, and it shows up in everything from architecture to video game design. Think about it: that turn you make? But here's the thing: most people learn the formula without understanding why it works, and that's where things fall apart.

An exterior angle is the angle formed between one side of a polygon and the extension of its adjacent side. For a regular polygon — one where all sides and angles are equal — every exterior angle is the same size. Still, if you imagine standing at a corner of a shape and walking along its edge, the exterior angle is the amount you rotate at that corner to continue along the next side. For an irregular polygon, they can all be different.

So how do you actually find it? That's what this post is about.

Why It Matters / Why People Care

You might be wondering why anyone needs to calculate exterior angles outside of a math class. In practice, this concept shows up more than you'd think.

For one, it's foundational in polygon geometry. If you know the exterior angles, you can figure out the interior angles, the number of sides, and whether a shape is regular or not. Engineers and designers use these relationships constantly — when they're tiling a floor, framing a structure, or even plotting a path around a building.

There's also a beautiful theorem at the center of all this: the sum of the exterior angles of any convex polygon is always 360 degrees, no matter how many sides it has. So naturally, a triangle, a hexagon, a 42-sided shape — they all add up to 360°. That fact is both elegant and incredibly useful, because it gives you a reliable shortcut no matter the polygon.

On a more practical level, if you're working with regular polygons — think stop signs (octagons), floor tiles (squares or hexagons), or nuts and bolts (hexagons again) — knowing the exterior angle helps you figure out how much material you need or how pieces fit together.

How It Works (or How to Do It)

Let's get into the mechanics. That said, finding an exterior angle isn't hard once you know which approach to use. The method depends on whether your polygon is regular or irregular, and on what information you already have.

The Basic Formula for Regular Polygons

For a regular polygon with n sides, each exterior angle is simply:

Exterior Angle = 360° ÷ n

That's it. Worth adding: if you have a regular hexagon (6 sides), each exterior angle is 360° ÷ 6 = 60°. For a regular decagon (10 sides), it's 360° ÷ 10 = 36°.

This works because the exterior angles, taken one at each vertex, always sum to 360°. In a regular polygon, they're all equal, so you just divide evenly.

Here's a quick mental check: as the number of sides increases, the exterior angle gets smaller. In real terms, 00036° each. A polygon with a million sides would have tiny exterior angles of 0.That tracks — a shape with a million sides is practically a circle, and the "turn" at each point is almost nothing.

Finding Exterior Angles in Irregular Polygons

For an irregular polygon, the exterior angles aren't all the same, so the formula above doesn't apply directly. But the sum still equals 360° — that never changes for a convex polygon.

If you know some of the exterior angles, you can find the missing ones by subtraction. Take this: if a convex quadrilateral has three known exterior angles of 80°, 100°, and 70°, the fourth is 360° − (80° + 100° + 70°) = 110°.

Want to learn more? We recommend which pair of elements has the most similar lewis structures and is the number 0 a rational number for further reading.

In practice, you'll often encounter a problem where you're given the interior angle and need the exterior one. That's where the next section comes in.

The Relationship Between Interior and Exterior Angles

At any vertex of a polygon, the interior angle and the exterior angle are supplementary — they add up to 180°. They sit on a straight line together.

So if you know the interior angle, you can find the exterior angle with:

Exterior Angle = 180° − Interior Angle

And vice versa:

Interior Angle = 180° − Exterior Angle

This is especially handy for regular polygons. If you know the interior angle of a regular polygon is 140°, the exterior angle is 180° − 140° = 40°. You can then confirm the number of sides: 360° ÷ 40° = 9, so it's a nonagon.

For irregular polygons, you might need to calculate interior angles first using other methods — like the interior angle sum formula, which is (n − 2) × 180° — and then subtract from 180° at each vertex to get the exterior angles individually.

A Step-by-Step Walkthrough

Let's walk through a concrete example so this all clicks.

Say you have a regular octagon (8 sides) and you want to find each exterior angle.

1. Identify the number of sides. An octagon has 8.2. Apply the formula. 360° ÷ 8 = 45°.
2. Verify. Multiply the exterior angle by the number of sides: 45° × 8 = 360°. It checks out.
3. Find the interior angle if needed. 180° − 45° = 135°.

Now try an irregular case. You have a convex pentagon with four known exterior angles: 50°, 70°, 90°, and 60°. The fifth exterior angle is 360° − (50° + 70° + 90° + 60°) = 360° −

360° − 270° = 90°. Worth adding: this ensures the total remains 360°, even with varying angles. For irregular polygons, if you’re given interior angles, subtract each from 180° to find their corresponding exterior angles. On top of that, for example, if an irregular quadrilateral has interior angles of 120°, 100°, 80°, and 60°, their exterior angles would be 60°, 80°, 100°, and 120°, respectively, summing to 360°. In problems where only some interior angles are known, calculate the missing exterior angles using the 180° rule, then use the total sum to solve for unknowns. To give you an idea, if a pentagon has interior angles of 150°, 130°, 110°, 90°, and an unknown angle, first find the fifth interior angle: (5 − 2) × 180° = 540° total interior angles. Subtract known angles: 540° − (150° + 130° + 110° + 90°) = 60°. In real terms, the exterior angle here is 180° − 60° = 120°. Which means always verify that all exterior angles sum to 360°, regardless of regularity. In real terms, this method ensures accuracy in both regular and irregular cases, reinforcing the consistency of polygon angle properties. ### Conclusion Understanding exterior angles provides a powerful tool for analyzing polygons. Whether regular or irregular, the sum of exterior angles remains 360°, a principle rooted in the geometry of convex shapes. By leveraging formulas like 360° ÷ n for regular polygons and the 180° supplementary relationship with interior angles, you can solve for unknowns efficiently. This knowledge not only simplifies calculations but also deepens your grasp of polygonal structures, from simple triangles to complex irregular forms. Mastery of these concepts is essential for tackling advanced geometry problems and appreciating the elegance of angular relationships in mathematics.