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
- 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 ).
- Substitute the number of sides
Since we are dealing with a regular octagon, we can substitute ( n = 8 ) into the formula.
- 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.
- Finish the apothem calculation
Substituting ( s ) (the length of a side of the octagon) into the apothem formula, we have:
$$ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $$
- 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)} ).
