Hexagon Calculator
The hexagon calculator finds side, area, perimeter, radius, diagonal and diameter of a regular hexagon. The calculator explains step-by-step how to find the unknown element.
Tutorial
What is a hexagon?
A regular hexagon has six equal angles and six equal sides. The measure of each angle is 1200. A hexagon has 9 diagonals; three long diagonals pass through the center.
Most important formulas?
The calculator uses the following formulas to find the missing elements of a hexagon.
1. Area
A regular hexagon consists of 6 equilateral triangles. The area of a sing triangle is a2√3/4, so the total area is
A = 6 × a2√3/4 = (3a2√3)/2
2. Perimeter
Since a hexagon has six equal sides the perimeter is
A = 6 × a
3. Incircle radius
The incircle radius is equal to the height of a single equilateral triangle so,
r = (a √3)/2
4. Circumcircle radius
The circumcircle radius is equal to the side of a single equilateral triangle so,
R = a
5. Long diagonal
d = 2a
6. Short diagonal
s = a√3
Solved examples
Example 1:
Find the area of a regular hexagon having a side equal to 6 cm.
In the example we will use formula for area of a regular hexagon:
A = (3a2√3)/2
A = (3·62√3)/2
A = (3· 36√3)/2
A = 3·18√3
A = 54√3
Example 2:
What is the side of the regular hexagon whose short diagonal is s = 12 cm?
In the example we will use short diagonal formula:
s = a × √3
12 = a × √3
a = 12/√3
a = 4√3
Example 3:
Find the side of a regular hexagon, if the area is 24√3.
Again we will start with the formula for the area
A = (3a2√3)/2
24√3 = (3a2√3)/2
2 × 24√3 = 3a^2√3
48√3 = 3a2√3
48 = 3a2
a2 = 12
a = √12
a = 2√3
Find the side $ a $ of a hexagon if incircle radius $r = 3\, \text{cm}$.
solution
$$ a = 2 \sqrt{ 3 }\, \text{cm} $$explanation
To find side $ a $ use formula:
$$ r = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$After substituting $r = 3\, \text{cm}$ we have:
$$ 3\, \text{cm} = \dfrac{ \sqrt{ 3 } \cdot a }{ 2 } $$ $$ \sqrt{ 3 } \cdot a = 3\, \text{cm} \cdot 2 $$ $$ \sqrt{ 3 } \cdot a = 6\, \text{cm} $$ $$ a = \dfrac{ 6\, \text{cm} }{ \sqrt{ 3 } } $$ $$ a = 2 \sqrt{ 3 }\, \text{cm} $$