Question

the ratio between the number of sides of two regular polygons is 3:4 and the ratio between the sum of their interior angles is 2:3. find the number of sides in each polygon.

Answer 1

let
s1 = the sum of the angles of the first polygon.
s2 = the sum of the angles of the second polygon.
n1 = the number of sides of the first polygon.
n2 = the number of sides of the second polygon.

the sum of the angles of a polygon is equal to 180 * (n-2)

for the first polygon,

s1 = 180 * (n1 - 2)

for the second polygon,

s2 = 180 * (n2 - 2)

the ratio of the sum of the angles of the polygons is 2/3.

this means that

s1/s2 = 2/3

  s1 = (2 * s2) / 3.

since

s1 is equal to 180 * (n1 - 2),

and

since s2 is equal to 180 * (n2 - 2),

 substitute in the equation for s1 to get:

s1 = (2 * s2) / 3 becomes:

180 * (n1 - 2) = (2 * 180 * (n2 - 2)) / 3.

divide both sides of this equation by 180

(n1 - 2) = (2 * 180 * (n2 - 2)) / (3 * 180).

since 180 in the numerator and denominator cancel out, the equation becomes:

(n1 - 2) = (2 * (n2 - 2)) / 3.

add 2 to both sides of this equation to get:

n1 = (2 * (n2 - 2)) / 3 + 2.

since 2 is equal to 6/3, the equation becomes:

n1 = (2 * (n2 - 2)) / 3 + 6/3.

you can now consolidate under the common denominator to get:

n1 = (2 * (n2 - 2) + 6) /3.

simplify by expanding the distribution to get:

n1 = (2 * n2 - 4 + 6) / 3.

combine like terms to get:

n1 = (2 * n2 + 2) / 3.

you are given that the ratio of n1/n2 = 3/4

from that ratio, you can solve for n1 to get:

n1 = (3 * n2) / 4


you can now replace n2 with its equivalent to get:

n1 = (2 * n2 + 2) / 3 becomes:

(3 * n2) / 4 = (2 * n2 + 2) / 3.

multiply both sides of this equation by 12 to get:

3 * (3 * n2) = (2 * n2 + 2) * 4.

simplify to get:

9 * n2 = 8 * n2 + 8

subtract 8 * n2 from both sides of this equation to get:

n2 = 8

since the ratio of n1 / n2 = 3/4, then the equation of:

n1 / n2 = 3/4 becomes:

n1 / 8 = 3/4

multiply both sides of this equation by 8 to get:

n1 = (3 * 8) / 4 which becomes:

n1 = 24 / 4 which becomes:

n1 = 6

you now have:

n1 = 6
n2 = 8

the ratio of n1/n2 is equal to 6/8 which is equal to 3/4.

s1 = 180 * (n1 - 2) = 180 * (6 - 2) = 180 * 4 = 720
s2 = 180 * (n2 - 2) = 180 * (8 - 2) = 180 * 6 = 1080

the ratio of s1/s2 is equal to 720/1080 which is equal to 2/3.

the ratio of the number of sides of the two polygons is equal to 3/4 as indicated in the problem statement.

the ratio of the sum of the angles of the two polygons is equal to 2/3 as indicated in the problem statement.

the solution to the problem is that the smaller polygon has 6 sides and the larger polygon has 8 sides.

Answer 2

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