swerve wrote:If a circle, regular hexagon, and a regular octagon have the same area and if the perimeter of the circle is represented by "c", that of the hexagon by "h" and that of the octagon by "o", then which of the following is true?
A. c > o > h
B. c > h > o
C. h > c > o
D. o > h > c
E. h > o > c
The OA is E.
Please, can anyone explain this PS question for me? I tried to solve it but I can't get the correct answer. Thanks.
This question becomes much easier if one remembers a general rule:
For a given perimeter length, a circle is the shape with the greatest area.
Therefore, for a given area, a circle is the shape with the least perimeter.
As the number of sides in a polygon increases, the shape of the polygon approaches that of a circle.
An octagon (8 sides) more resembles a circle than does a hexagon (6 sides).
Therefore, if a circle, hexagon and octagon are equal in area, the octagon's perimeter will be shorter than that of the hexagon and longer than that of the circle.
The correct answer is choice
E.
In case you don't remember the rule above, the question can also be answered by Plugging In.
For each shape, find the ratio of its perimeter to its area by Plugging In a value for its perimeter.
Any Plug In values will work, but here I'll set the perimeter of each shape equal to 1.
For the circle, Area = π(r)^2 and Perimeter = 2π(r).
Perimeter = 1 = 2Ï€(r).
Simplify to find the circle's radius, r = 1/(2Ï€).
Find the circle's area, A = π/[(2π)^2] = 1/(4π).
Thus, the ratio of the circle's perimeter to its area is
4Ï€.
The perimeter of the hexagon is also 1, so each of the hexagon's six sides has a length of 1/6.
Bisect the hexagon three times by connecting opposite vertices, dividing the hexagon into 6 equal triangles.
At the center of the hexagon, six equal angles sum to 360°, so each triangle has a 60° angle at the center of the hexagon.
At the perimeter of the hexagon, the interior angles sum to (n - 2)180 degrees (where n = the number of sides), so there are 720° in a hexagon.
Since the hexagon is regular, each of its six interior angles measures 720° ÷ 6 = 120°.
And since each of the 120° angles is bisected by a diagonal, the hexagon consists of six equilateral triangles.
Bisecting each of the six equilateral triangles produces twelve 30-60-90 triangles. Use the properties of this special right triangle to determine the area of hexagon.
The ratio of the lengths of the sides in a 30-60-90 triangle is x : x√3 : 2x, respectively.
The base of each 30-60-90 triangle is 1/12 (half of 1/6), so the height of each 30-60-90 triangle √3/12.
The area of a triangle is half the product of its base and height, so the area of each of the twelve triangles is (1/2)*(1/12)*(√3/12).
And since the hexagon consists of twelve such triangles, the area of the hexagon is 12*(1/2)*(1/12)*(√3/12) = √3/24.
Thus, the ratio of the hexagon's perimeter to its area is
24/√3.
Compare the ratio of the circle's perimeter to its area and the ratio of the hexagon's perimeter to its area.
24/√3 is greater than 4π (√3 is roughly equal to 1.7, and π is roughly equal to 3.1), so this ratio is greater in the hexagon.
If the areas of the hexagon and circle are equal, then the perimeter of the hexagon is longer than the circumference of the circle.
An octagon resembles a circle more closely than does a hexagon, so the perimeter of the octagon will be greater than the perimeter of the circle and less than the perimeter of the hexagon.
The correct answer is choice
E. $$$$
