There are two polygons. The larger one has three times as many sides as the smaller one. Its angle sum is four times as big. How many sides does the smaller polygon have?
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Q: There are two polygons. The larger one has three times as many sides as the smaller one. Its angle sum is four times as big. How many sides does the smaller polygon have? (Need full solution)?
“Three times as many sides” means the larger polygon has a multiple of 3 sides. 3 is the fewest sides any polygon can have, so it has not less than 9 sides.
Its angle sum is four times the smaller one’s. This applies to interior angles.
A triangle’s sum is 180. A square’s is 360. A pentagon’s is 540. Any sum will be a multiple of 180, and equal to the number of sides minus two, then multiplied by 180.
A hexagon has 6 sides, so 4×180=720 degrees sum. That part fits the triangle, but we know that can’t be the answer because 3×3=9.
So start at 9.
7×180÷4 can’t be a multiple of 180. What we need is a multiple of 3 which is 2 greater than a multiple of 4.
18 fits that description.
The 18-gon’s interior sum is 16×180=2880, divide that by 4 to get 720, which is the hexagon’s sum.
18÷6=3, so we have a solution. The smaller polygon can be a hexagon.
We should check the next one though. Multiples of 3: 21, 24, 27, 30, 33, 36, 39, 42 (every 4th multiple of 3 will be 2 more than a multiple of 4).
So the 30-gon:
28×180=5040
5040÷4=1260
1260÷180=7
7+2=9
9×3=27
Off by 3.
The 42-gon:
((((((42–2)×180)÷4)÷180)+2)×3)=36
Off by 6. It will keep getting farther away, so we have a unique solution.