Using trigonometric ratios to express area of regular polygonsI am very confused. My book just asked me to use trigonometric ratios to express the area of a regular polygon with 9 sides and lengths of 8.So far I have learned how to use the sin cos and tan in right triangles and have no idea how this applies to all polygons. Can someone please explain this to me

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Olivia Petersen · Accepted Answer

Step 1Sketch a regular 9-gon. Join the centre of the 9-gon to the 9 vertices. This divides the 9-gon into 9 congruent triangles. To find the area of the 9-gon, we will find the area of one of these triangles, and multiply the result by 9.Concentrate now on one of these triangles AOB, where O is the centre and A and B are two consecutive vertices. We will find the area of   △  A  O  B.Step 2We have   A  B  =  8, and   ∠  A  O  B  =            360      ∘        9    =      40    ∘  .Drop a perpendicular from O to AB, meeting AB at M. Note that   ∠  A  O  M  =      20    ∘  .Also,             A      M              O      M        =  tan  ⁡  (      20    ∘    ). But   A  M  =  4, so   O  M  =      4          tan      ⁡      (              20        ∘            )      .Now we know the base AB of   △  A  O  B, and the height OM. The area of   △  A  O  M is therefore       1    2    ⋅  8  ⋅      4          tan      ⁡      (              20        ∘            )      .Finally, multiply by 9 and simplify a bit. We get       144          tan      ⁡      (              20        ∘            )      , or if you prefer,   144  tan  ⁡  (      70    ∘    )

musicbachv7 · Accepted Answer

Step 1If the polygon has 9 sides, the angle of each of the 9 sectors is             2      π        9  .If you cut this angle in half and draw an altitude to one of the sides, you get a right triangle with base 4 (half of 8) and an angle of             2      π        18   (the angle is also cut in half). So the area of that piece is base*height/2 or             1      2        (  4  )  (  4  )  cot  ⁡      (                  2        π            18        )  Step 2The area of the whole polygon is 18 times that, so   A  =  8  ⋅  18  cot  ⁡      (                  2        π            18        )   and I&#039;m bad at mental math so I&#039;ll let you finish simplifying.

Use trigonometric ratios to express the area of a regular polygon with 9 sides and lengths of 8.

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