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Regular Polygons

Lesson · Beginner

Geometry: Plane Geometry

For an n-sided polygon,

Sum of interior angles=180∘(n−2)\text{Sum of interior angles}=180^\circ(n-2)

Regular Hexagon

For a regular hexagon with side length a:

a)

∠A=180∘(6−2)6=120∘\angle A=\frac{180^\circ(6-2)}{6}=120^\circ

b)

Regular hexagon divided into six equilateral triangles
[ABCDEF]=6(a234)=3a232[ABCDEF]=6\left(\frac{a^2\sqrt{3}}{4}\right)=\frac{3a^2\sqrt{3}}{2}

c)

Short diagonal AC of a regular hexagon
AC=a3AC=a\sqrt{3}

d)

Long diagonal of a regular hexagon
AD=2aAD=2a

Regular Octagon

For a regular octagon with side length a:

a)

∠A=180∘(8−2)8=135∘\angle A=\frac{180^\circ(8-2)}{8}=135^\circ

b)

Regular octagon inside a square
[ABCDEFGH]=(a+a2)2−4(a2/22)[ABCDEFGH]=(a+a\sqrt{2})^2-4\left(\frac{a^2/2}{2}\right)
=a2+2a2+2a22−a2=2a2(1+2)=a^2+2a^2+2a^2\sqrt{2}-a^2=2a^2(1+\sqrt{2})

c)

Short diagonal AC of a regular octagon
AC=a2+2AC=a\sqrt{2+\sqrt{2}}
AC2=a2+a2−2a2cos⁡135∘=2a2+2a222AC^2=a^2+a^2-2a^2\cos135^\circ=2a^2+2a^2\frac{\sqrt{2}}{2}
=2a2+a22=a2(2+2)=2a^2+a^2\sqrt{2}=a^2(2+\sqrt{2})

d)

Diagonal AD of a regular octagon
AD=a2+a=a(2+1)AD=a\sqrt{2}+a=a(\sqrt{2}+1)

e)

Long diagonal AE of a regular octagon
AE=a4+22AE=a\sqrt{4+2\sqrt{2}}
AE2=a2+AD2=a2+(a(2+1))2AE^2=a^2+AD^2=a^2+\left(a(\sqrt{2}+1)\right)^2
=a2+a2(3+22)=a2(4+22)=a^2+a^2(3+2\sqrt{2})=a^2(4+2\sqrt{2})
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