3.4 Polygons


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3.4 Contents


3.4.1 Universal Properties

Closed planar objects with at least three straight sides and equal number of vertices (corners)
Perimeter Sum of side lengths
Area Sum of areas of triangles divided within shape
Sum of Angles $(\text{number of sides}-2)·\pi$

3.4.2 Types

Equilateral
Equiangular
Regular
Irregular
Convex & Concave

3.4.3 Regular Polygons

Jump to Circles
Perimeter $n·s$
Area $n·s·h/2$
Sum of angles $(n-2)·\pi$
Angle $∡(h,r)$ $\pi/n$
Circumradius $h·\sec \Big(\frac{\pi}{n}\Big) \land \frac{s}{2}·\csc \Big(\frac{\pi}{n}\Big)$
Area using apothem $n·h^2·\tan \Big(\frac{\pi}{n}\Big)$
Area using side $\frac{n·s^2}{4}·\cot \Big(\frac{\pi}{n}\Big)$
Area using circumradius $n·r^2·\sin \Big(\frac{\pi}{n}\Big)·\cos \Big(\frac{\pi}{n}\Big)$
Proof of Circumradius
With respect to the center angle between the apothem and circumradius, use the corresponding right angle definitions for $h$ = adjacent, $b/2$ = opposite, and $r$ = hypotenuse, and solve for $r$ in each case $$\sec \Big(\frac{\pi}{n}\Big)=\frac{r}{h}\qquad\csc \Big(\frac{\pi}{n}\Big)=\frac{r}{s/2}$$
Proof of Area using Apothem
Set the two circumradius formulae equal to each other $$h·\sec \Big(\frac{\pi}{n}\Big)=\frac{s}{2}·\csc \Big(\frac{\pi}{n}\Big)$$ Multiply by $h$, then divide by the cosecant function $$h^2·\frac{\sec(\pi/n)}{\csc(\pi/n)}=\frac{s·h}{2}$$ Use the right angle definitions of the secant and cosecant functions to simplify into the tangent function $$\frac{\sec(\pi/n)}{\csc(\pi/n)}=\frac{r/h}{r/s}=\frac{s}{h}=\tan \Big(\frac{\pi}{n}\Big)$$ Substitute $$h^2·\tan \Big(\frac{\pi}{n}\Big)=\frac{s·h}{2}$$ Multiply by the number of sides $$n·h^2·\tan \Big(\frac{\pi}{n}\Big)=n·\frac{s·h}{2}$$
Proof of Area using Circumradius
Using the first circumradius equality, isolate $h$ $$h=\frac{r}{\sec(\pi/n)}$$ Substitute for $h$ in the equation for the area using the apothem $$A=n·r^2·\frac{\tan(\pi/n)}{\sec^2(\pi/n)}$$ Use the right angle definitions of the tangent and secant functions to simplify to sine and cosine $$\frac{\tan(\pi/n)}{\sec^2(\pi/n)}=\frac{s/h}{r^2/h^2}$$ $$=\frac{s}{r}·\frac{h}{r}=\sin\Big(\frac{\pi}{n}\Big)·\cos\Big(\frac{\pi}{n}\Big)$$ Substitute $$A=n·r^2·\sin\Big(\frac{\pi}{n}\Big)·\cos\Big(\frac{\pi}{n}\Big)$$
Proof of Area using Side Length
Substitute the second circumradius equality into the area using the circumradius equation, then expand $$A=\frac{n·s^2}{4}·\csc^2\Big(\frac{\pi}{n}\Big)·\sin\Big(\frac{\pi}{n}\Big)·\cos\Big(\frac{\pi}{n}\Big)$$ Use the right angle definitions to simplify the trig functions into cotangent, $$\csc^2\Big(\frac{\pi}{n}\Big)·\sin\Big(\frac{\pi}{n}\Big)·\cos\Big(\frac{\pi}{n}\Big)$$ $$=\frac{r^2}{s^2}·\frac{s}{r}·\frac{h}{r}=\frac{h}{s}=\cot\Big(\frac{\pi}{n}\Big)$$ Substitute $$A=\frac{n·s^2}{4}·\cot\Big(\frac{\pi}{n}\Big)$$

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