Hexagon Shape: Sides, Angles And Area
The short version
- A hexagon is a closed, flat shape with six straight sides, six corners and six interior angles.
- The interior angles of any hexagon add up to 720 degrees. In a regular hexagon they’re all equal, so each one is 720 ÷ 6 = 120 degrees.
- Perimeter of a regular hexagon: 6 × side. Area of a regular hexagon: (3√3 ÷ 2) × side², which is about 2.598 × side².
- A regular hexagon splits cleanly into six equilateral triangles, which is where the area formula comes from and why hexagons tile a floor with no gaps.
A hexagon is a polygon with six straight sides and six angles, and its interior angles always add to 720 degrees. If all six sides are the same length and all six angles are equal, it’s a regular hexagon and every angle measures 120 degrees. If the sides or angles differ, it’s an irregular hexagon, and the 720 degree total still holds.
The name comes from Greek: “hex” for six and “gonia” for angle. A stop sign is not a hexagon (eight sides makes it an octagon), but a honeycomb cell is, and so is the head of most bolts.
What makes a shape a hexagon
Three things have to be true: the shape is closed with no openings, every side is a straight segment, and there are exactly six of them. Nothing about being pretty or symmetrical. A long, lopsided six sided figure is as much a hexagon as the tidy one in the textbook.
Hexagons also have 9 diagonals, from the polygon formula n(n – 3) ÷ 2. With n = 6 that’s 6 × 3 ÷ 2 = 9.
Regular and irregular hexagons
| Type | What’s true | Angle sum |
|---|---|---|
| Regular hexagon | All six sides equal, all six angles equal at 120 degrees | 720 degrees |
| Irregular hexagon | Sides or angles differ; angles can be anything that adds correctly | 720 degrees |
| Convex hexagon | Every interior angle is less than 180 degrees; no corner caves in | 720 degrees |
| Concave hexagon | At least one interior angle is more than 180 degrees, so a corner points inward | 720 degrees |
These overlap. A regular hexagon is also convex, and an L shaped hexagon is irregular and concave at once. The area formula below only works on the regular kind, which is the biggest trap in hexagon homework.
Hexagon angles and where 720 comes from
Pick any vertex of a hexagon and draw diagonals to every vertex you can reach. You get four triangles. Each triangle’s angles add to 180 degrees, and together those triangles cover the whole hexagon with nothing left over.
4 × 180 = 720 degrees.
The general rule for any polygon with n sides is (n – 2) × 180. For a hexagon, n = 6, so (6 – 2) × 180 = 720. This works for irregular hexagons too, which is what makes missing angle problems solvable.
Worked example. Five angles of a hexagon measure 110, 125, 140, 95 and 130 degrees. Find the sixth.
110 + 125 + 140 + 95 + 130 = 600
720 – 600 = 120 degrees.
Exterior angles are simpler. They add to 360 degrees in any polygon, so each one in a regular hexagon is 360 ÷ 6 = 60 degrees. Check it: 120 + 60 = 180, the angle of a straight line. If that pairing is new, Supplementary Angles, Explained covers it.
Perimeter and area of a hexagon
Perimeter
Add all six sides. For a regular hexagon that shortcuts to P = 6s.
A regular hexagon with 7 cm sides has perimeter 6 × 7 = 42 cm. Going the other way, a regular hexagon with a 48 cm perimeter has sides of 48 ÷ 6 = 8 cm.
Area from the side length
For a regular hexagon with side s:
Area = (3√3 ÷ 2) × s², and since √3 is about 1.732, the multiplier works out to about 2.598.
Take a regular hexagon with 4 inch sides.
s² = 4 × 4 = 16
Area = 2.598 × 16 = 41.57 square inches.
Why that odd multiplier? Draw lines from the center of a regular hexagon to each vertex and you cut it into six identical equilateral triangles, each with side s. One equilateral triangle has area (√3 ÷ 4) × s², and six of them give (6√3 ÷ 4) × s², which simplifies to (3√3 ÷ 2) × s². Same formula, now with a reason behind it.
Checking the 4 inch hexagon: one triangle is (1.732 ÷ 4) × 16 = 6.93 square inches, and 6 × 6.93 = 41.57. Matches.
Area from the apothem
The apothem runs from the center out to the middle of a side. Any regular polygon uses this formula:
Area = 1/2 × perimeter × apothem
For a regular hexagon the apothem equals (√3 ÷ 2) × s, which is about 0.866 × s. With 4 inch sides:
apothem = 0.866 × 4 = 3.46 inches
perimeter = 24 inches
Area = 0.5 × 24 × 3.46 = 41.57 square inches.
Same answer. Use whichever number the problem gives you.
Area of an irregular hexagon
No formula covers these, so cut the shape into rectangles and triangles and add. Picture an L shaped hexagon with sides of 8, 5, 3, 2, 5 and 3 units and a right angle at every corner.
It fits inside an 8 by 5 rectangle with a 5 by 2 rectangle missing from the top left.
8 × 5 = 40
5 × 2 = 10
40 – 10 = 30 square units.
Its perimeter is 8 + 5 + 3 + 2 + 5 + 3 = 26 units, the same as the 8 by 5 rectangle it came from. Cutting a rectangular notch from a corner changes the area but never the perimeter.
Why hexagons show up everywhere
Regular hexagons tile a surface with no gaps and no overlaps. Three of them meet at a point, and 3 × 120 = 360 degrees, which is exactly a full turn. Only triangles, squares and hexagons manage that on their own, and among those three the hexagon uses the least wall length to enclose the most area.
That’s why bees build hexagonal cells: less wax, more honey. The same shape shows up in basalt columns, graphite and graphene, snowflake symmetry, insect eyes, and the nut on a bolt, where six flat faces give a wrench plenty of grip angles.
Stretch a hexagon into the third dimension and you get a hexagonal prism: 8 faces, 18 edges, 12 vertices. An unsharpened pencil is the everyday version.
Common mistakes with hexagons
Calling a stop sign a hexagon. It has eight sides, so it’s an octagon. Count the sides before naming any polygon.
Using the regular hexagon area formula on an irregular one. (3√3 ÷ 2) × s² needs all six sides equal. On a lopsided hexagon it gives a number that means nothing.
Mixing up the apothem and the radius. The apothem reaches the middle of a side (about 0.866s). The radius reaches a vertex and equals s exactly. Swapping them throws the area off.
Multiplying 180 by 6. The angle sum is (n – 2) × 180, not n × 180. Six sides means four triangles, not six.
Forgetting to square the side. Writing 2.598 × 4 instead of 2.598 × 16 is the most common arithmetic slip in these problems. Area always involves squared units.
Practice problems
- How many sides, vertices and angles does a hexagon have?
- What is the sum of the interior angles of any hexagon?
- A regular hexagon has sides of 7 cm. What is its perimeter?
- Five interior angles of a hexagon measure 100, 130, 140, 115 and 125 degrees. Find the sixth angle.
- What is the measure of one exterior angle of a regular hexagon?
- Find the area of a regular hexagon with 6 inch sides, rounded to the nearest tenth.
- A regular hexagon has a perimeter of 48 cm. Find the side length and the distance from one vertex straight across to the opposite vertex.
- A hexagon has right angles at every corner and side lengths of 8, 5, 3, 2, 5 and 3 units going around. Find its perimeter and area.
Answers
1. Six of each. Six sides, six vertices, six interior angles.
2. 720 degrees, from (6 – 2) × 180 = 4 × 180.
3. 42 cm, since 6 × 7 = 42.
4. 110 degrees. The five given angles total 100 + 130 + 140 + 115 + 125 = 610, and 720 – 610 = 110.
5. 60 degrees, from 360 ÷ 6 = 60. It pairs with the 120 degree interior angle to make 180.
6. About 93.5 square inches. s² = 36, and 2.598 × 36 = 93.53, which rounds to 93.5.
7. Each side is 48 ÷ 6 = 8 cm. The long diagonal across a regular hexagon is twice the side, so 2 × 8 = 16 cm.
8. Perimeter 26 units, area 30 square units. The sides add to 8 + 5 + 3 + 2 + 5 + 3 = 26. The shape is an 8 by 5 rectangle (40 square units) with a 5 by 2 rectangle (10 square units) removed, and 40 – 10 = 30.
What it means if your child is stuck on this
Hexagon problems almost never fail because of the hexagon. They fail on the steps around it. A student who writes 2.598 × 4 instead of 2.598 × 16 has a squaring problem, not a geometry problem.
The other common blocker is the missing angle question, which is a one step equation in costume: 610 + x = 720. Kids who miss that connection treat every missing angle problem as a brand new puzzle.
The useful step is finding out which layer is shaky. Is My Child Behind In Math? 12 Signs By Grade Level covers what to look for, and Math Skills By Grade Level: What Your Child Should Know shows where shapes and area sit by grade.
Frequently asked questions
How many sides does a hexagon have?
Six. It also has six vertices and six interior angles, plus 9 diagonals connecting nonadjacent corners.
What are the angles of a hexagon?
The interior angles of any hexagon add to 720 degrees. In a regular hexagon each one is 120 degrees. In an irregular hexagon the individual angles vary, but they still total 720.
What is the area formula for a hexagon?
For a regular hexagon, Area = (3√3 ÷ 2) × s², or about 2.598 × s², where s is the side length. For an irregular hexagon there’s no single formula, so split it into rectangles and triangles and add the pieces.
Is a hexagon a regular polygon?
Only when all six sides and all six angles are equal. Plenty of hexagons are irregular. The six sided shape on a soccer ball panel is regular, but an L shaped six sided room is not.
Why do bees use hexagons?
Hexagons tile with no wasted space, and of the three shapes that tile on their own, the hexagon encloses the most area for the least perimeter. That means less wax used per cell of honey stored.
Find the actual gap
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