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How to find the apothem of an octagon?

Understand the Problem

The question is asking how to calculate the apothem of a regular octagon. The apothem can be found using geometric formulas, usually involving the length of the sides and the number of sides of the polygon.

Answer

$$ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $$
Answer for screen readers

The apothem ( a ) of a regular octagon is given by the formula:

$$ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $$

where ( s ) is the length of one side.

Steps to Solve

  1. Identify the formula for the apothem of a regular octagon

The apothem ( a ) of a regular polygon can be calculated using the formula:

$$ a = \frac{s}{2 \tan(\frac{\pi}{n}) $$

where ( s ) is the length of one side, and ( n ) is the number of sides. For an octagon, ( n = 8 ).

  1. Substitute the number of sides

Since we are dealing with a regular octagon, we can substitute ( n = 8 ) into the formula.

  1. Calculate ( \tan(\frac{\pi}{n}) )

Now we calculate the tangent of (\frac{\pi}{8}):

$$ \tan\left(\frac{\pi}{8}\right) $$

You can use a calculator or trigonometric tables to find this value.

  1. Finish the apothem calculation

Substituting ( s ) (the length of a side of the octagon) into the apothem formula, we have:

$$ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $$

  1. Simplify and Solve

Finally, perform the arithmetic calculation to find the value of ( a ) once you have the side length ( s ) and the calculated tangent.

The apothem ( a ) of a regular octagon is given by the formula:

$$ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $$

where ( s ) is the length of one side.

More Information

The apothem is an important segment in geometry as it represents the shortest distance from the center to a side of the polygon. It is particularly useful in calculating area and when working with inscribed circles.

Tips

  • Confusing the number of sides; make sure you use ( n = 8 ) for an octagon.
  • Forgetting to convert angles to radians if using a calculator; ensure it's set to the correct mode.
  • Not using the correct formula for the apothem; double-check that you are using ( a = \frac{s}{2 \tan\left(\frac{\pi}{n}\right)} ).
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