Sum Of Angles In A Hexagon

You're staring at a hexagon — maybe it's a tile pattern on a bathroom floor, a honeycomb cell, or a geometry problem that showed up on your kid's homework. Six corners. In real terms, six sides. And somewhere in the back of your mind, a question nags: what do all those angles actually add up to?

Most people guess 360. That's the circle number, the triangle number, the "safe" answer. But a hexagon isn't a triangle. It's not a quadrilateral either. And the answer isn't 360.

It's 720 degrees. Always. Every single hexagon, regular or lopsided, convex or dented inward — the interior angles sum to 720. No exceptions.

What Is the Sum of Angles in a Hexagon

Let's get the definition out of the way. A hexagon is any polygon with six straight sides and six vertices. In practice, that's it. The sides don't have to be equal. On top of that, the angles don't have to match. As long as it's a closed shape with six straight edges, it's a hexagon.

The interior angles* are the angles inside the shape at each vertex. Add all six together and you get 720 degrees. This holds for every hexagon that doesn't cross over itself — what mathematicians call a simple polygon.

Regular vs. Irregular Hexagons

A regular hexagon is the symmetric one you probably picture first: six equal sides, six equal angles. Each interior angle measures exactly 120 degrees. 120 × 6 = 720. Clean.

An irregular hexagon? Two could be acute, four obtuse. Also, doesn't matter. Practically speaking, the angles can be all over the place. Think about it: another 150. Day to day, one might be 90 degrees. Another 135. The total stays locked at 720.

Convex vs. Concave

Convex hexagons bulge outward everywhere — every interior angle is less than 180 degrees. Here's the thing — that dent doesn't change the sum. Concave hexagons have at least one "dent" where an interior angle exceeds 180 degrees (a reflex angle). The reflex angle just eats up more of the 720 budget, leaving less for the other five angles.

Exterior Angles

Walk around the outside of any hexagon, turning at each corner. The total turn you make — the sum of exterior angles — is always 360 degrees. This is true for every* simple polygon, triangle through n-gon. Each exterior angle pairs with its interior neighbor to make 180 degrees (a straight line). Six vertices × 180 = 1080. Subtract the 360 you turned walking around, and you're left with 720 inside.

Why It Matters / Why People Care

You might wonder: who actually needs this?

Tiling and Design

Hexagons tile the plane perfectly — no gaps, no overlaps. That's why bees build honeycombs that way. Think about it: it's why bathroom tiles, game boards (hello, Settlers of Catan), and graphene lattices all use hexagons. The 120-degree interior angle is the magic number: three hexagons meet at a point, 120 + 120 + 120 = 360. Flat. Seamless.

If you're designing a hexagonal patio, a quilt pattern, or a board game, you're working with that 720-degree total whether you realize it or not.

Engineering and Architecture

Hexagonal structures show up in bridge trusses, dome frameworks, and satellite components. The geometry distributes stress efficiently. Engineers calculating joint angles in a hexagonal truss need to know exactly how the interior angles partition — especially when the hexagon isn't regular.

Computer Graphics and Game Dev

Procedural generation of hexagonal grids, pathfinding on hex maps, collision detection for hex-shaped hitboxes — all of it leans on angle math. This leads to the 720-degree sum is a sanity check. If your generated hexagon's angles don't total 720, something's broken in the code.

Standardized Tests and Math Competitions

SAT, ACT, GRE, AMC, MathCounts — they all love polygon angle problems. Practically speaking, a classic: "Five angles of a hexagon measure 110°, 125°, 130°, 140°, and 155°. What's the sixth angle?" You can't solve it without knowing the total is 720.

How It Works (or How to Find the Sum)

When it comes to this, three ways stand out. Pick the one that clicks for you.

The Triangle Method (Most Intuitive)

Pick any vertex of a hexagon. Still, draw diagonals from that vertex to all other non-adjacent vertices. You'll draw three diagonals, splitting the hexagon into four triangles.

Each triangle's angles sum to 180 degrees. Four triangles × 180 = 720 degrees.

This works because the triangles' angles are the hexagon's angles — just partitioned differently. Practically speaking, no angle is lost or gained. The diagonals just slice the existing angles into smaller pieces that happen to form triangles.

The Formula: (n − 2) × 180

The triangle method generalizes instantly. Any n-sided polygon can be split into (n − 2) triangles from a single vertex. So the interior angle sum is:

Sum = (n − 2) × 180°

Want to learn more? We recommend lack of access to improved sanitation facilities in slums and find the area of the following parallelogram for further reading.

For a hexagon, n = 6: (6 − 2) × 180 = 4 × 180 = 720.

This formula is worth memorizing. It works for triangles (1 × 180 = 180), quadrilaterals (2 × 180 = 360), pentagons (3 × 180 = 540), heptagons (5 × 180 = 900), and so on.

The Exterior Angle Walk

Imagine standing at one vertex of a hexagon, facing along one side. Walk forward to the next vertex. Turn by the exterior angle. Consider this: walk the next side. Also, turn again. Repeat until you're back where you started, facing the original direction.

You've made one full revolution: 360 degrees of turning total.

At each vertex, interior angle + exterior angle = 180° (they form a straight line). Six vertices gives 6 × 180 = 1080° total of interior + exterior pairs.

Subtract the 360° you turned (the sum of exterior angles): 1080 − 360 = 720° interior.

This method is elegant because it proves the exterior angle sum is always* 360° for any simple polygon — and the interior sum follows automatically.

Finding a Missing Angle

This is the practical skill. You know five angles. You need the sixth.

Step 1: Add the five known angles. Step 2: Subtract from 720. Step 3: That's your missing angle.

Example: Known angles are 118°, 122°, 12

7°, 135°, and 143°. Adding these gives 545°. The missing angle is 720° − 545° = 175°.

Regular Hexagons: When All Angles Are Equal

What if your hexagon is regular — all sides equal, all angles equal?

Divide the total sum by 6: 720° ÷ 6 = 120° per angle.

Each interior angle in a regular hexagon measures exactly 120 degrees. This is why honeycomb cells work so efficiently — nature found the optimal tiling angle.

Beyond Hexagons: The General Pattern

The formula scales beautifully. For any polygon with n sides:

• Triangle (n=3): 180°
• Quadrilateral (n=4): 360°
• Pentagon (n=5): 540°
• Hexagon (n=6): 720°
• Heptagon (n=7): 900°
• Octagon (n=8): 1080°

Each additional side adds 180° to the total. This linear relationship reveals something profound: polygons are just triangles wearing disguises.

Convex vs. Concave: Does It Matter?

Here's the surprising part: the 720° sum holds for concave hexagons too. A "dented" hexagon still has interior angles totaling 720° — some angles just measure more than 180°.

The catch? Also, you can't use the triangle method directly on a concave polygon. But the formula (n−2) × 180° remains unshaken.

Why This Matters for Real Problems

Polygon angle sums aren't just academic exercises. They're the foundation for:

• Computer graphics: Calculating surface normals for 3D models
• Architecture: Ensuring structural angles add up correctly
• Engineering: Designing gears, wheels, and mechanical linkages
• Game development: Implementing collision detection with polygonal hitboxes

The SAT doesn't ask you to calculate 720° because they want you to memorize it. They ask because they want you to understand that 720° is inevitable — a mathematical truth that cannot be violated.

Quick Reference

• Hexagon interior angle sum: 720°
• Regular hexagon single angle: 120°
• Hexagon exterior angle sum: 360°
• Formula for any n-gon: (n − 2) × 180°

The next time you see a stop sign, remember: its eight 135° angles total exactly 1080°, derived from the same principle that gives hexagons their 720°. Mathematics isn't scattered facts — it's a connected web where each strand supports the whole.