# the internal angle, and the sum of the internal angles in any N-sided polygon?

We have:

triangles have $3\times 60°=180°$

squares have $4\times 90°=360°$

pentagon have $5\times 108°= 540°$

hexagons have $6\times 120°=720°$

heptagons have $7\times 128.57° = 899.99 = 900°$

octagons have $8\times 135°=1080°$

What is the sum of the internal angles $\Pi$ in any $N$-sided polygons?

What is the value of one of the angle $\alpha$ in any $N$-sided polygons?

What is the value of $\Pi/\alpha$ in any $N$-sided polygons?

what is the value of $(\Pi-\alpha)/\alpha$ in any $N$-sided polygons?

i can draw the geometry. • the answer to the first is Π=(n−2)⋅180∘ and the answer to the third is Π/α=N² but there is something between the both. – user52413 Aug 4 '13 at 12:41
• Hi you have changed the question . $\alpha$ is external angle . you did not mention it previously . – Harish Kayarohanam Aug 4 '13 at 16:05
• Note : external angle = $360^{\circ} -internal angle$ – Harish Kayarohanam Aug 4 '13 at 16:06
• i gave you a point to help me asking the question. It isn't coming from a book. – user52413 Aug 4 '13 at 16:11
• Even you did not mention $\Pi$ is an internal angle and $\alpha$ is an external angle . Any way after knowing internal angle formula its easy to go further . – Harish Kayarohanam Aug 4 '13 at 16:13

$$\mbox{sum of internal angles of polygon with N sides } = (N-2) * \pi$$

from which you can deduce the rest

$$\mbox{each internal angle \alpha} = \frac{(N-2) * \pi}{N}$$

$$\pi/\alpha = \frac{N}{N-2}$$

$$\frac{\pi - \alpha}{\alpha} = \frac{2}{N-2}$$

$$\frac{sum\ of\ internal\ angles}{each\ angle\ \alpha} = N$$ $$\frac{sum\ of\ internal\ angles - each\ angle\ \alpha}{each\ angle\ \alpha} = N -1$$ where '$N$' is the number of sides of polygon and

$\pi = 180^{\circ}$

Note: in your question you have used $\Pi$ to denote something. both cases if it denotes $180 ^{\circ}$ or if it denotes sum of internal angles , have been solved in my answer . I don't want to use $\Pi$ again (as i have already used it to denote $180 ^{\circ}$ ) and confuse readers . Hope this helps .

• @Daniel Rust . Thanks for the edit . I was searching for this degree symbol .But what does "circ" in \circ means . Is it circle ? – Harish Kayarohanam Aug 4 '13 at 12:48
• I'm not sure if it's standard for degrees but it's the usual symbol used for function composition (eg $f\circ g (x)=f(g(x))$) and it works well as an index to be a degree symbol. The circ refers to circle yes. – Dan Rust Aug 4 '13 at 12:58
• thanks for edit. for example for (π−α)/α i have π=7 * 128.57° = 899.99 = 900° and (π−α)/α=7*771,43/128,57 =42 – user52413 Aug 4 '13 at 13:01
• @user52413 , what are u suggesting ? – Harish Kayarohanam Aug 4 '13 at 13:04
• α=((N−2)∗π)/N but π is not factor of N – user52413 Aug 4 '13 at 13:11