hexagon connecting the centers of the outer circles

Ans is -- 6 sqrt3

Hexagon can be viewed as 6 equilateral triangles with each side as dia of the circle... 2.... so area of 1 such triangle is sqrt 3/4 *2 square... which is sqrt 3 and 6 is 6sqrt 3... the answer... hope this helps.

r=1, therefore each side of the hexagon is 2. You can also split the hexagon into 2 half and you have two trapezoids with a base of 2 and 4.

You can also determine the size of each of the angle on the trapezoid by using the formula;

[(number of sides -2)180]/6 = [(6-2)180] = (4*180)/6 = 720/6 = 120

You need to know this so you can calculate the height of the trapezoid. Draw a line from the corner of the smaller base straight down to form a 90 degree angle with the larger base. This creates a 30-60-90 triangle because you have 90 degree angle where you drew the line down and a 60 degree angle where you split the trapezoid in two. From there you can deduce that the other angle must be 30 degrees.

So you have a 30-60-90 triangle with a hypotenuse of 2. 30-60-90 triangles sides are in the ratio x : x[sqrt(3)] : 2x, from smallest side to largest side. You hypotenuse is 2 so the smallest side of the triangle must be 1 and the middle length side must be 1[sqrt(3)]. Thus, the height of the trapezoid is sqrt(3).

So, formula for the area of a trapezoid is;

[(base 1 +base 2)*height]/2, so

[(4+2)*sqrt(3)]/2 = (6sqrt(3))/2

But we have two trapezoids in the circle so the area of the hexagon is;

[(6sqrt(3))/2]*2 = 6sqrt(3)

M811 wrote:


Seven identical circles - each of radius 1 inch - are arranged tangentially to one another as shown above. What is the area enclosed by the hexagon connecting the centers of the outer circles?
3
6
6\/2
6\/3
Cannot be determined

S=3/2*(3^1/2)*R^2, where S area hexagon. and R radius of circumscribed circle
here R=1+1=2
S=3/2*3^1/2*2^2=6*3^1/2
my answer D

The hexagon can be divided into 6 equilateral triangles of length 2 inch each.
Area of an equilateral triangle = (\/3) /4 * (side) ^ 2
area of the hexagon = 6 * [\/3 /4 * (side) ^ 2 ] = 6\/3

M811 wrote:


Seven identical circles - each of radius 1 inch - are arranged tangentially to one another as shown above. What is the area enclosed by the hexagon connecting the centers of the outer circles?
3
6
6\/2
6\/3
Cannot be determined