Hexagons: properties, angles and symmetry
Categories: gcse geometry polygon
Level:
A hexagon is a polygon (a closed, flat shape with straight sides) with 6 sides and 6 vertices. Hexagons can be regular or irregular, depending on the sides and angles. Here are some example hexagons:
It is easy to spot a hexagon. If a shape is a polygon and it has 6 sides and 6 corners, it is a hexagon.
The name hexagon comes from Ancient Greek. Hex means six, and gonia means corner or angle. So a hexagon is a shape with six corners. Hexagons can also be called 6-gons, but that term is not used very often.
Want to learn more about GCSE geometry? Explore our guide to geometry, including angles, polygons, transformations and other important topics.
Hexagon quick facts
| Property | Value |
|---|---|
| Number of sides | 6 |
| Number of vertices | 6 |
| Interior angle sum | 720° |
| Regular interior angle | 120° |
| Regular exterior angle | 60° |
| Number of diagonals | 9 |
| Lines of symmetry (regular) | 6 |
| Rotational symmetry order (regular) | 6 |
Regular and irregular hexagons
A regular hexagon is a 6-sided shape where every side is the same length and every corner has the same angle. All regular hexagons have the same shape. They can be different sizes, and they can be drawn at an angle, which might make them look slightly different, but these two shapes are both regular hexagons:
An irregular hexagon is a 6-sided shape where the sides are not all equal, or the angles are not all equal. Irregular hexagons come in many shapes. Here are some examples:
A regular hexagon is made from 6 equilateral triangles
A regular hexagon can be divided into 6 equilateral triangles of equal size, like this:
Every side of every triangle has length s, which is also equal to the length of the sides of the hexagon. Every angle of every triangle is 60°, because all equilateral triangles have angles of 60°. This helps calculate the angles in a regular hexagon, as we will see later.
If you want to know why the triangles are equilateral, see the proof later in this article.
Interior angles of a hexagon
The interior angles of a hexagon are shown here:
The interior and exterior angles lesson gives this formula for the sum of interior angles:
$$ \mathit{sum} = (n - 2) \times 180^\circ $$
Where n is the number of sides. In this case, the number of sides n is 6, so the sum of the interior angles is:
$$ \begin{aligned} \mathit{sum} &= (6 - 2) \times 180^\circ \\ &= 4 \times 180^\circ \\ &= 720^\circ \end{aligned} $$
Interior angle of a regular hexagon
For a regular hexagon, all the interior angles are equal:
We know from the previous section that the sum of all the interior angles in any hexagon is 720°. But for a regular hexagon, we also know that all the interior angles are equal. So we can find the angle at each corner by dividing the sum of the angles by 6:
$$ \mathit{angle} = \frac{720^\circ}{6} = 120^\circ $$
For regular hexagons, there is another, simpler way to find the angle. We can divide the hexagon into 6 equilateral triangles, each with interior angles of 60°:
Each vertex of the hexagon includes two triangle vertices, each of 60°. So the total interior angle is:
$$ \mathit{angle} = 60^\circ + 60^\circ = 120^\circ $$
Exterior angles of a hexagon
The exterior angles of a hexagon are shown here:
The interior and exterior angles lesson tells us that the sum of the exterior angles of any polygon is 360°.
For a regular hexagon, all the exterior angles are equal:
We know that the sum of the exterior angles of any polygon is 360°, and there are 6 equal angles, so each angle is:
$$ \mathit{angle} = \frac{360^\circ}{6} = 60^\circ $$
Again, there is another, simpler method for finding the exterior angles of a regular hexagon. This diagram shows the hexagon, divided into 6 equilateral triangles:
At point P, there are three angles on a line:
- The exterior angle x
- Two angles that are both interior angles of an equilateral triangle, so each angle is 60°.
We know that the angles on a straight line add up to 180°, so:
$$ x + 60^\circ + 60^\circ = 180^\circ $$
This means that the exterior angle x must be 60°.
Symmetry of a regular hexagon
A regular hexagon has 6 lines of symmetry. This means that it can be reflected over each of the 6 grey lines shown here:
A regular hexagon has rotational symmetry of order 6. This means that it fits onto itself six times during a complete 360° rotation. Each time it rotates by 60° (a 6th of a turn), it will fit into the original shape.
These rules only apply to regular hexagons. An irregular hexagon might have fewer lines of symmetry and a lower order of rotational symmetry. Many irregular hexagons have no symmetry at all.
Diagonals of a hexagon
A hexagon has 9 diagonals. A diagonal is a straight line joining two vertices that are not next to each other.
In a regular hexagon, 3 of the diagonals pass through the centre. Those are the diagonals that join opposite vertices.
The other 6 diagonals do not pass through the centre. They join vertices that are separated by one vertex.
Perimeter of a regular hexagon
We can find the perimeter of a regular hexagon with sides of length s. The perimeter is the sum of the side lengths. For a regular hexagon, all the sides are the same length s, and a hexagon has 6 sides, so the perimeter is:
$$ \mathit{perimeter} = 6 \times s $$
Area of a regular hexagon
What is the area of a regular hexagon with side length s? An easy way to calculate the area is to remember that a regular hexagon is made up of 6 equilateral triangles of side s.
The triangle area lesson tells us that the area of an equilateral triangle with side length s is:
$$ \mathit{triangle\ area} = \frac{\sqrt{3}}{4}s^2 $$
Since a regular hexagon is made up of 6 equilateral triangles, its area is:
$$ \mathit{hexagon\ area} = 6\frac{\sqrt{3}}{4}s^2 = \frac{3\sqrt{3}}{2}s^2 $$
Hexagons in everyday life
Regular hexagons tessellate (fit together perfectly with no gaps), making them a popular shape for floor and bathroom tiles. The only other regular polygons that tessellate are equilateral triangles and squares.
Other common examples of hexagons in real life include:
 |
Honeycomb - bees use hexagonal cells because the shape tessellates perfectly and uses the least wax for a given area. |
 |
Hex bolts and nuts - the hexagonal cross-section allows a spanner to grip from multiple angles. |
 |
Pencils - many pencils have a hexagonal cross-section, so they don't roll off desks. |
 |
Snowflakes are roughly hexagonal - they usually have six lines of symmetry because ice crystals form a hexagonal structure. |
Why are regular hexagons made from equilateral triangles?
A regular hexagon can be divided into 6 triangles by drawing a line from the centre to each vertex, as shown in part 1 below:
Earlier, we said that the triangles are equilateral. Here is the proof:
- We know that a regular hexagon has rotational symmetry of order 6. This means that if we rotate the shape by one-sixth of a turn, it maps onto the original shape exactly.
- This can only happen if all six triangles are identical (in maths, we say they are congruent).
- This means that all six triangles must have the same angle A (otherwise they would not be identical).
- Since six angles A make up a full turn (360°), angle A must equal 60°.
- Also notice that, when we rotate the shape through 60°, side b maps exactly onto side c. Therefore, sides b and c must be the same length. This is shown in part 2 of the diagram above.
- If b and c are the same, the triangle must be an isosceles triangle. This means that angles B and C must also be identical.
- Angles A + B + C must add up to 180°. We know that A is 60°, and we also know that B = C, so B and C must also equal 60°. This is shown in part 3 of the diagram.
- The triangle has all three angles equal to 60°, therefore it is an equilateral triangle.
Common mistakes
Here are some common mistakes that people make when working with hexagons:
- Don't assume every hexagon is regular, unless the question tells you that it is.
- Don't assume every interior angle of a hexagon is 120°. That is true for regular hexagons but might not be true for irregular hexagons.
- Don't confuse the sum of the interior angles (720°) with the size of each interior angle (120° in a regular hexagon).
- Don't use 180° instead of 360° when calculating an exterior angle.
- Don't forget to divide by six when calculating an individual angle in a regular hexagon.
- Don't confuse the number of diagonals with the number of sides.
Worked example 1
A hexagon has five interior angles measuring:
110°, 125°, 130°, 115° and 120°.
Find the sixth angle.
**Step 1: Calculate the sum of the interior angles. **
A hexagon has six sides, so:
(6 − 2) × 180° = 720°
**Step 2: Add the five known angles. **
110° + 125° + 130° + 115° + 120° = 600°
**Step 3: Subtract from the total. **
720° − 600° = 120°
**Answer: The missing angle is 120°. **
Practice questions
Q1 What is the sum of the interior angles of a hexagon?
Q2 Calculate the size of each exterior angle of a regular hexagon.
Q3 A regular hexagon has a perimeter of 54 cm. Calculate the length of each side.
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