geometry:regular polygons

baiju09 wrote:this is from an old book

The ratio between the number of sides of two regular polygons is 1:2 and the ratio between their interior angles is 2:3. Which of the following is the number of sides of these polygons, respectively?
a. 4,8
b. 5,10
c. 6,12
d. 7,14
e. 8,16

If n and 2 n are the number of sides in those polygons, and 2 x and 3 x were their interior angles, respectively, then each exterior angle of those polygons will be

180 – 2 x and 180 – 3 x respectively.

Hence, 360/n = 180 – 2 x and 360/ (2 n) = 180 – 3 x; solving we get [spoiler]n = 4.

A[/spoiler]

Easier here is to remember each interior angle of an n-sided regular polygon is given by 180 (n – 2)/n, and check the choices…

@A: n = 4 in 180 (n – 2)/n = 90; and n = 8 in 180 (n – 2)/n = 135, ratio 90:135 is yes 2:3.

[spoiler]A[/spoiler]

checking the answer you gave gives us a quadrilateral and an octagon, 4 sides and 8 sides, which should have 360 and 1080 as interior angles. doesn't work. in fact, as written, I don't think there is a solution to the problem that is positive, provided the rule for interior angles of a polygon follows the (n-2)(180) formula, where n is the number of sides

Claremont32 wrote:checking the answer you gave gives us a quadrilateral and an octagon, 4 sides and 8 sides, which should have 360 and 1080 as interior angles. doesn't work. in fact, as written, I don't think there is a solution to the problem that is positive, provided the rule for interior angles of a polygon follows the (n-2)(180) formula, where n is the number of sides

@Claremont32-
The interior angle for the 4 sided regular polygon is 360/4=90 and that of a 8 sided polygon is 1080/8 = 135

So the ratio is 90/135 = 2/3

Claremont32 wrote:checking the answer you gave gives us a quadrilateral and an octagon, 4 sides and 8 sides, which should have 360 and 1080 as interior angles. doesn't work. in fact, as written, I don't think there is a solution to the problem that is positive, provided the rule for interior angles of a polygon follows the (n-2)(180) formula, where n is the number of sides

Hi Claremont,

It seems you are considering the ratio of the sum of the interior angles and not the ratio of the interior angles. 180(n-2) gives the sum of the interior angles. If you consider the ratio to be 180(n1-2):180(n2-2)::2:3 then your answer will be negative.

However, what we should be considering is 180(n-2)/n which the value of each interior angle. You should clearly get n1=4 and n2 = 8(since, n2=2n1). In fact, we can straight ways consider the ratio between 180(n1-2)/n1 and 180(n2-2)/n2 instead of considering that each interior angle is (180- each external angle) while each exterior angle is 360/n and this a1 = 180 - 360/n1 and a2 = 180 - 360/n2.

Hope that helps!

Anirban