Given the radius of the circle inscribing the hexagon is #r# express the shaded area in terms of #r#?

Question

Noah Atkinson · Accepted Answer

Given the radius of the circle inscribing the hexagon is r

The area of the circle #Delta_c=pir^2#

The area of the each of 6 equilateral triangles having length of each side r is #Delta_t=sqrt3/4r^2#

Area of each of the segment of circle marked X
#=Delta_x=1/6Delta_c-Delta_t#

From the figure it is obvious that the area of Yellow shaded region in the given figure is

#Delta_y=2Delta_t-4xxDelta_x#

#=>Delta_y=2Delta_t-4xx(1/6Delta_c-Delta_t)#

#=>Delta_y=6Delta_t-2/3Delta_c#

#=>Delta_y=6xxsqrt3/4r^2-2/3xxpir^2#

#=>Delta_y=(3sqrt3)/2r^2-2/3xxpir^2#

#=>Delta_y=((9sqrt3-4pi))/6r^2#

Avery Allen · Answer

#0.5036* r^2#

#OA=OC=OD=OB=OE=OF=r#
# angleOCD=360^@/6=60^@ => Delta OCD# is equilateral
Area #DeltaOCD=sqrt3/4*r^2#

The 3 green areas in sector #OCD# are the same.

One green area #A_G=pir^2*60/360-sqrt3/4*r^2#
#=> A_G=(2pi-3sqrt3)/12*r^2#

The black area #A_B# in sector #OCD = pir^2*60/360- 3A_G#
#=> A_B=(pir^2)/6-3((2pi-3sqrt3)/12)*r^2#
#=((2pi-6pi+9sqrt3)/12)*r^2#
#=((9sqrt3-4pi)/12)*r^2#

Now, let the shaded area in your diagram be #A_S#
#=> A_S=2*A_B#
#=> A_S=2*((9sqrt3-4pi)/12)*r^2#

#=> A_S=((9sqrt3-4pi)/6)*r^2=0.5036*r^2#

Evelyn Arboleda · Answer

undefined The shaded area of the inscribed hexagon can be expressed in terms of the radius $ r $ of the circle as follows:

$$ \text{Shaded Area} = 6 \times \left( \frac{1}{2} r \right)^2 \times \tan\left(\frac{\pi}{6}\right) $$

or simplified as:

$$ \text{Shaded Area} = \frac{3\sqrt{3}r^2}{2} $$

This formula is derived from the fact that the interior angles of a regular hexagon are each $ \frac{\pi}{3} $ radians, and the apothem (distance from the center of the hexagon to the midpoint of one of its sides) is equal to $ \frac{1}{2}r $. Therefore, the area of each triangular sector formed by two radii and one side of the hexagon can be calculated using the formula for the area of a triangle, which is $ \frac{1}{2} \times \text{base} \times \text{height} $, where the base is the length of one side of the hexagon (equal to $ r $) and the height is the apothem (equal to $ \frac{1}{2}r $). Finally, since there are 6 identical triangular sectors in the hexagon, we multiply the area of one sector by 6 to get the total shaded area.