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pappueshwar
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hi all,
this is a very good problem . request to assist in understanding the same:
Regular hexagon ABCDEF has a perimeter of 36. O is the center of the hexagon and of circle O. Circles A, B, C, D, E, and F have centers at A, B, C, D, E, and F, respectively. If each circle is tangent to the two circles adjacent to it and to circle O, what is the area of the shaded region (inside the hexagon but outside the circles)?
explanation given by manhattan:Shaded area = Area of the hexagon - (area of circle O) - (portion of circles A, B, C, D, E, F that is in the hexagon)
With a perimeter of 36, the hexagon has a side that measures 6. The regular hexagon is comprised of six identical equilateral triangles, each with a side measuring 6. We can find the area of the hexagon by finding the area of the equilateral triangles.
The height of an equilateral triangle splits the triangle into two 30-60-90 triangles (Each 30-60-90 triangle has sides in the ratio of 1: root 3 : 2). Because of this, the area for an equilateral triangle can be expressed in terms of one side. If we call the side of the equilateral triangle, s, the height must be (s)root3 / 2 (using the 30-60-90 relationships).
The area of a triangle = 1/2 × base × height, so the area of an equilateral triangle can be expressed as: 1/2 × s × (s) root3 / 2 = 1/2 × 6 × 3 root (3) = 9root 3
Area of hexagon ABCDEF = 6 × 9 = 54 root 3
For circles A, B, C, D, E, and F to have centers on the vertices of the hexagon and to be tangent to one another, the circles must be the same size. Their radii must be equal to half of the side of the hexagon, 3. For circle O to be tangent to the other six circles, it too must have a radius of 3.
Area of circle O = r2 = 9pi
To find the portion of circles A, B, C, D, E, and F that is inside the hexagon, we must consider the angles of the regular hexagon. A regular hexagon has external angles of 360/6 = 60°, so it has internal angles of 180 - 60 = 120°. This means that each circle has 120/360 or 1/3 of its area inside the hexagon.
The area of circles A, B, C, D, E, and F inside the hexagon = 1/3(9) × 6 circles = 18.
Thus, the shaded area = 54 root 3 - 9pi - 18pi = 54root 3 - 27pi . The correct answer is E.
my doubt is :
A) unable to understand how height of the triangle is calculated using 30-60-90 rule.
B) A regular hexagon has external angles of 360/6 = 60°, so it has internal angles of 180 - 60 = 120°. what are these external angles and internal angles. how are they arrived..is this something to be remembered and mugged up?
this is a very good problem . request to assist in understanding the same:
Regular hexagon ABCDEF has a perimeter of 36. O is the center of the hexagon and of circle O. Circles A, B, C, D, E, and F have centers at A, B, C, D, E, and F, respectively. If each circle is tangent to the two circles adjacent to it and to circle O, what is the area of the shaded region (inside the hexagon but outside the circles)?
explanation given by manhattan:Shaded area = Area of the hexagon - (area of circle O) - (portion of circles A, B, C, D, E, F that is in the hexagon)
With a perimeter of 36, the hexagon has a side that measures 6. The regular hexagon is comprised of six identical equilateral triangles, each with a side measuring 6. We can find the area of the hexagon by finding the area of the equilateral triangles.
The height of an equilateral triangle splits the triangle into two 30-60-90 triangles (Each 30-60-90 triangle has sides in the ratio of 1: root 3 : 2). Because of this, the area for an equilateral triangle can be expressed in terms of one side. If we call the side of the equilateral triangle, s, the height must be (s)root3 / 2 (using the 30-60-90 relationships).
The area of a triangle = 1/2 × base × height, so the area of an equilateral triangle can be expressed as: 1/2 × s × (s) root3 / 2 = 1/2 × 6 × 3 root (3) = 9root 3
Area of hexagon ABCDEF = 6 × 9 = 54 root 3
For circles A, B, C, D, E, and F to have centers on the vertices of the hexagon and to be tangent to one another, the circles must be the same size. Their radii must be equal to half of the side of the hexagon, 3. For circle O to be tangent to the other six circles, it too must have a radius of 3.
Area of circle O = r2 = 9pi
To find the portion of circles A, B, C, D, E, and F that is inside the hexagon, we must consider the angles of the regular hexagon. A regular hexagon has external angles of 360/6 = 60°, so it has internal angles of 180 - 60 = 120°. This means that each circle has 120/360 or 1/3 of its area inside the hexagon.
The area of circles A, B, C, D, E, and F inside the hexagon = 1/3(9) × 6 circles = 18.
Thus, the shaded area = 54 root 3 - 9pi - 18pi = 54root 3 - 27pi . The correct answer is E.
my doubt is :
A) unable to understand how height of the triangle is calculated using 30-60-90 rule.
B) A regular hexagon has external angles of 360/6 = 60°, so it has internal angles of 180 - 60 = 120°. what are these external angles and internal angles. how are they arrived..is this something to be remembered and mugged up?
