Let's say we have hexagon ABCDEF, where those letters are the vertex (corner) points of the hexagon.
AB = 6 is one side length of the hexagon. Because it's a regular hexagon, the other side lengths are 6 cm as well.
Let G be the center of the hexagon. You can find the center by intersecting two perpendicular bisectors of two of the sides of the hexagon.
Point G is set up such that we can form equilateral triangle ABG. All sides of this triangle are the same length. The nice thing about regular hexagons is that they are composed of exactly 6 congruent equilateral triangles. We'll only need to focus on one such triangle.
For triangle ABG, we know,
AB = 6
BG = AB = 6
AG = AB = 6
or we could say
AB = 6
BG = 6
AG = 6
This is enough to show that the radius of the circumscribed circle is 6 cm (BG and AG are radii of the circumscribed circle).
This is shown in the diagram below.
The circumscribed circle is in red.
The circumscribed circle is the smallest circle that wraps around the hexagon.
No part of the hexagon spills outside the red circle.
All vertex points of the hexagon are on the circumscribed circle.
With this radius value in mind, we can compute the circumference of the red circle.
C = 2*pi*r
C = 2*pi*6
C = 12pi
The circumference of the circumscribed circle is exactly 12pi centimeters.
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The blue circle is the inscribed circle. It's the largest possible circle contained entirely inside the hexagon. Think of trying to pack a cylindrical vase (the base being a circle) inside a hexagonal container.
The diagram shows that the radius of the blue circle is GH
Triangle GBH is a right triangle with HB being half as long as the hexagon's side. So HB = 3.
Use the pythagorean theorem to get
a^2+b^2 = c^2
(GH)^2+(HB)^2 = (GB)^2
(GH)^2+(3)^2 = (6)^2
(GH)^2+9 = 36
(GH)^2 = 36-9
(GH)^2 = 27
GH = sqrt(27)
GH = sqrt(9*3)
GH = sqrt(9)*sqrt(3)
GH = 3*sqrt(3)
The radius of the inscribed circle is exactly 3*sqrt(3) cm
Now we can find the circumference of the blue circle
C = 2*pi*r
C = 2*pi*3*sqrt(3)
C = 6pi*sqrt(3)
The circumference of the inscribed circle is exactly 6pi*sqrt(3) centimeters.
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Answers:
Circumference of the circumscribed circle = 12pi
Circumference of the inscribed circle = 6pi*sqrt(3)
Values are exact and the units for each are in cm.
Use your calculator to compute the approximate values if needed.