Question

Please help... Each interior angle of a regular polygon is p times each exterior angle. Find an expression in terms of p for (1) an exterior angle (2) the number of sides of the polygon

Answer 1

Let each interior angle be A
And each exterior angle be B
It's given that each interior angle is p time the exterior angle.
\[A=p\times B\]

We know for a regular polygon
exterior angle + interior angle=180
so
\[A+B=180\]
A=pB
so
\[pB+B=180\]
or
\[B(p+1)=180\]
or
exterior angle B
\[B=\frac{180}{p+1}\]
First part solved

Answer 2

Got it! Thank you! :) What about second part?

Answer 3

Now interior angle A for a regular polygon with n sides
\[A=(n-2)\frac{180}{n}\]
so
\[nA=(n-2) \times 180\]
or
\[nA-180n=-360\]
or
\[n(A-180)=-360\]
or
\[n=\frac{360}{180-A}\]
180-A=B
\[n=\frac{360}{B}\]
and pB=A
or B=A/p
so
\[n=\frac{360p}{A}\]

Answer 4

Thank you so much for answering this! :)