A regular hexagon is inscribed in a circle with a radius
of 10 in. What is the area of the shaded region rounded
to the nearest square inch?

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Mathematics

Find the area of the regular hexagon if the radius of a circle inscribed in the hexagon is 10√3 meters.

Step-by-step answer

P Answered by Specialist

D. 600√3

Step-by-step explanation:

Refer to your previous question:

Radius of the inscribed circle is the apothem of the hexagon.

Apothem (a) and half of the side (s) make a 30-60-90 right triangle.

The ratio of the legs, as per property of 30-60-90 triangle:

s : a = 1 : √3 ⇒ s : 10√3 = 1 : √3 ⇒ s = 10

Half the side is s = 10 units, then side of the hexagon is 20 units.

The area of the hexagon:

A = 1/2Pa, P- perimeter, a- apothem A = 1/2(6*20)*(10√3) = 600√3

Correct choice is D

Mathematics

Find the area of the regular hexagon if the radius of a circle inscribed in the hexagon is 10√3 meters.

Step-by-step answer

P Answered by Master

D. 600√3

Step-by-step explanation:

Refer to your previous question:

Radius of the inscribed circle is the apothem of the hexagon.

Apothem (a) and half of the side (s) make a 30-60-90 right triangle.

The ratio of the legs, as per property of 30-60-90 triangle:

s : a = 1 : √3 ⇒ s : 10√3 = 1 : √3 ⇒ s = 10

Half the side is s = 10 units, then side of the hexagon is 20 units.

The area of the hexagon:

A = 1/2Pa, P- perimeter, a- apothem A = 1/2(6*20)*(10√3) = 600√3

Correct choice is D

Mathematics

What radius of a circle is required to inscribe a regular hexagon with an area of 210.44 cm2 and an apothem of 7.794 cm

Step-by-step answer

P Answered by PhD

We know that

[area of a regular hexagon]=6*[area of one equilateral triangle]
210.44=6*[area of one equilateral triangle]
[area of one equilateral triangle]=210.44/6> 35.07 cm²

[area of one equilateral triangle]=b*h/2
h=7.794 cm
b=2*area/h> b=2*35.07/7.794>b= 9 cm

the length side of a regular hexagon is 9 cm
applying the Pythagorean theorem
r²=h²+(b/2)²>r²=7.794²+(4.5)²> r²=81> r=9 cm

this last step was not necessary because the radius is equal to the hexagon side> (remember the equilateral triangles)

the answer is
the radius is 9 cm

Mathematics

What radius of a circle is required to inscribe a regular hexagon with an area of 210.44 cm2 and an apothem of 7.794 cm

Step-by-step answer

P Answered by PhD

Mathematics

What radius of a circle is required to inscribe a regular hexagon with an area of 210.44 cm2 and an apothem of 7.794 cm?

Step-by-step answer

P Answered by PhD

$\bf \textit{area of a regular polygon}\\\\
A=\cfrac{1}{2}ap\quad 
\begin{cases}
a=apothem\\
p=perimeter\\
-------\\
a=7.794\\
A=210.44
\end{cases}\implies 210.44=\cfrac{1}{2}(7.794)p
\\\\\\
420.88=7.794p\implies \cfrac{420.88}{7.794}=p\implies 54\approx p
\\\\\\
\textit{a \underline{hexa}gon has 6 sides, so }\cfrac{54}{6}\implies \stackrel{\textit{length of one side}}{9}$

so.. now, if we know one side is of 9 units long... ok, bear in mind that a regular hexagon, splits the center of the circumscribing circle in 6 even angles, that'd be 60° each, if you run a bisector from that angle, it'd split into two 30° angles, check the picture below.

So it ends with a 30-60-90 triangle with one side being just half of the 9 units, and we can just use the 30-60-90 rule to get the radius itself, which is just one of the sides of the 30-60-90 triangle.

What radius of a circle is required to inscribe a regular hexagon with an area of 210.44 cm2 and an