A circle is inscribed within a regular hexagon in such a way that the

Question

A circle is inscribed within a regular hexagon in such a way that the circle touches all sides of the hexagon at exactly one point per side. Another circle is drawn to connect all the vertices of the hexagon. Expressed as a fraction, what is the ratio of the area of the smaller circle to the area of the larger circle?

A. √(2/3)
B. (√2)/3
C. (√3)/2
D. (√3)/4
E. 3/4

Kudos for a correct solution.

A. √(2/3)
B. (√2)/3
C. (√3)/2
D. (√3)/4
E. 3/4

Kudos for a correct  solution .

Bunuel · Accepted Answer

MANHATTAN GMAT OFFICIAL SOLUTION: The first thing to recognize is that a regular hexagon has 6 equal sides and 6 equal internal angles of 120°, since the angles inside the hexagon must add up to (n – 2)×180 = 720°, or 120° per angle. This is a very symmetrical figure. The circle will touch each side exactly in the middle of the side, by symmetry, like so: hex-1.jpg The outer circle will touch each vertex: hex-2.jpg Now, to compare the areas of the two circles, we should compare their radii. The obvious place to draw radii is from the points of contact with the circles: hex-3.jpg The triangle that’s been created is a 30-60-90 triangle. At point B, the radius is perpendicular to the side of the hexagon (which is tangent to the circle). The 120° angle of the hexagon is equally split by the longer radius, again by symmetry, creating a 60° angle at point A. The ratios of the sides of a 30-60-90 triangle are 1: √3 : 2, with 2 as the longest side (the hypotenuse). The longer radius is the “2” side, while the shorter radius is the “√3” side. Since the areas are proportional to the square of the radius (by A = πr^2), we know that the smaller area to the larger area would be \sqrt{(3)^2} : 2^2 , or 3 : 4. Expressed as a fraction, this ratio is 3/4. Note that C is a trap answer: it expresses the ratios of the radii themselves. The correct answer is E.

rombo27 · Answer

given x the radius of the circle external to hexagon. The area of the external circle is X^2pi

Each angle of the hexagon is 120° . The hexagon is composed of 6 equilateral triangles. Each hight is the radius of the inner circle. The height is also the hight of a right triangle with angles 60°,90°, 30° . So, given the formula, the height/radius is (x/2)SQR3.

The area of the inner circle is ((x/2)SQR3)^2Pi = (¾)x^2Pi
The area of the external circle is x^2Pi

Hence the ratio is ((¾)x^2Pi ) / x^2Pi = ¾ Answer E.

peachfuzz · Answer

A hexagon has 6 sides. 
360/6=60

We know that there are 6 even central angles, 60 degrees each.
Splicing a 60 degree central angle in half, we have a 30-60-90 triangle. 
The hypotenuse of that triangle (the radius of the large circle) is in the ratio of 2, and the height of the triangle is in the ratio of sqrt(3).

sqrt(3)^2 *pi / 2^2 * pi
= 3/4

Answer: E

madzaka · Answer

Easier, faster and safer method than this 30-60-90 that everyone does

You have the 2 circles inside and outside with radius to the center of the side of the hexagon and to the edge.

You take one of the 6 parts to make triangle and split it in half.
(Here everyone talks about 30-60-90 but it is too slow)

you do pythagor theorem (equation 1) you get everything on power of 2nd which is relatable to the areas cuz they have power on 2nd

second equation is that you get an Equilateral triangle for the 1 of the 6th parts and here all sides are the same and you can get the relation between the side on the power of 2nd to the half side on the power of second (eq.2)

substitute 2 in 1 and you have your relation already with powers.

then R1^2/R2^2 comes out immedietly (Area 1 / Area 2 = pi*R1^2 / pi*R2^2)

bumpbot · Answer

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A circle is inscribed within a regular hexagon in such a way that the

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