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655-705 Level|   Geometry|      
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A circle is inscribed in a regular hexagon, what is ratio of the area 02 May 2020, 07:37
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58% (02:20) correct 42% (02:02) wrong based on 67 sessions

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A circle is inscribed in a regular hexagon, what is ratio of the area of the hexagon to the area of the circle?


A. \(2√2 : π\)

B. \(3√2 : π\)

C. \(2√3 : π\)

D. \(3√3 : π\)

E. \(4√3 : π\)


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C
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A circle is inscribed in a regular hexagon, what is ratio of the area 02 May 2020, 08:03
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Bunuel
A circle is inscribed in a regular hexagon, what is ratio of the area of the hexagon to the area of the circle?


A. \(2√2 : π\)

B. \(3√2 : π\)

C. \(2√3 : π\)

D. \(3√3 : π\)

E. \(4√3 : π\)

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exterior angle = 360/6 = 60
interior angle = 180 -60 = 120
Attachment:
img1.png
img1.png [ 27.22 KiB | Viewed 9430 times ]
radius of the circle will be √3a/2.

area(hexagon)/area(circle) = \(\frac{3√3a^2*4}{2*\pi * (√3a)^2}\) = \(\frac{3√3a^2*4}{2*\pi * 3*a^2}\)= \(\frac{2√3}{\pi }\)
Ans: C
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A circle is inscribed in a regular hexagon, what is ratio of the area 02 May 2020, 08:05
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Bunuel
A circle is inscribed in a regular hexagon, what is ratio of the area of the hexagon to the area of the circle?


A. \(2√2 : π\)

B. \(3√2 : π\)

C. \(2√3 : π\)

D. \(3√3 : π\)

E. \(4√3 : π\)


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Please the solution in the screenshot as well as Video solution stating each step of the solution

Answer: Option C




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Screenshot 2020-05-02 at 8.31.00 PM.png [ 878.6 KiB | Viewed 9089 times ]
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A circle is inscribed in a regular hexagon, what is ratio of the area 07 Jan 2024, 00:07
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A hexagon is a regular polygon.

(1) The sum of interior angles of a polygon is given by \((n-2)*180^{\circ} \), where \(n\) is the number of sides. For a regular polygon, each interior angle would be equal and thus the measure of each interior angles would be equal to \(\frac{(n-2)*180^{\circ} }{ n}\).

(2) The central angle of a regular polygon is given by \(\frac{360 }{ n}\), which is basically \(\frac{360}{6}\) = \(60^{\circ}\).

From (1) and (2) above we can see that the angle formed at O is \(60^{\circ}\), while the angles formed at the vertices E and D are \(120^{\circ}\), which are further bisected and each is then divided into two angles of \(60^{\circ}\).

Thus, we get an equilateral triangle (since each angle is \(60^{\circ}\)), which means that all the sides of the triangles are equal to each other. Let's assume that each side of the hexagon is s, the other two sides of the equilateral triangle then are also s each. The height of the equilateral triangle is equal to \(\frac{\sqrt{3} }{ 2} * s\).

The height of the equilateral triangle is equal to the radius of the circle, since the circle touches all the sides of the hexagon (the largest such circle possible).

Thus area of circle = \((\frac{\sqrt{3}}{2} *s)^2 * \pi\)
Area of polygon = \(\frac{1}{2} * apothem * perimeter = \frac{1}{2} * (\frac{\sqrt{3}}{2} *s) * 6s \)

Apothem: The apothem (sometimes abbreviated as apo) of a regular polygon is a line segment from the center to the midpoint of one of its sides. Equivalently, it is the line drawn from the center of the polygon that is perpendicular to one of its sides.

Thus, the ratio of the areas = \(\frac{3 \sqrt{3}(s)^2 }{2} : (\frac{3}{4} *s) * \pi = 2 \sqrt{3} : \pi\)
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apothem-regular-polygon.png
apothem-regular-polygon.png [ 71.48 KiB | Viewed 3841 times ]

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