How many sides has a regular polygon if the measure of one of its angles is 128 4/7º? 144º? 160º?
For a regular polygon, let "n" be the number of sides and "\\alpha" be the interior angle.
The sum of interior angles for a regular polygon is "n\\alpha" .
Also we can use another formula for calculating the sum of interior angles: "180^\\circ (n-2)" .
"180^\\circ (n-2)=n\\alpha\n\\\\\n180^\\circ n-360^\\circ =n\\alpha \n\\\\\nn(180^\\circ -\\alpha )=360^\\circ \n\\\\\nn=\\frac{360^\\circ}{180^\\circ -\\alpha }"
1) "\\alpha =128 \\ 4\/7 ^\\circ"
"n=\\frac{360^\\circ}{180^\\circ -128\\ 4\/7^\\circ }=\\frac{360^\\circ}{51\\ 3\/7^\\circ}=7"
2) "\\alpha =144 ^\\circ"
"n=\\frac{360^\\circ}{180^\\circ -144^\\circ }=\\frac{360^\\circ}{36^\\circ}=10"
3) "\\alpha =160^\\circ"
"n=\\frac{360^\\circ}{180^\\circ -160^\\circ }=\\frac{360^\\circ}{20^\\circ}=18"
Answer: 7 sides; 10 sides; 18 sides.
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