How to find the apothem of an octagon?
Understand the Problem
The question is asking for the method to find the apothem of a regular octagon, which is the line from the center to the midpoint of one of its sides. Typically, this involves knowing the length of the sides or the radius of the circumscribed circle.
Answer
The apothem can be calculated using $ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $ or $ a = R \cos\left(\frac{\pi}{8}\right) $.
Answer for screen readers
The apothem of a regular octagon can be calculated using the formula $ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $ or $ a = R \cos\left(\frac{\pi}{8}\right) $, depending on whether you know the side length or the circumradius.
Steps to Solve
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Identify Variables To find the apothem ($a$) of a regular octagon, you can use the formula involving the length of a side ($s$) or the radius of the circumscribed circle ($R$).
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Formula Using Side Length The apothem can be calculated using the side length of the octagon with the following formula: $$ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $$ This gives us a method to find the apothem if we know the side length.
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Formula Using Circumradius If you have the radius of the circumscribed circle ($R$), you can find the apothem using: $$ a = R \cos\left(\frac{\pi}{8}\right) $$ This method uses the radius directly to calculate the apothem.
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Calculate the Apothem Choose the appropriate formula based on the information you have (either side length or circumradius) and plug in the value. For example, if $s = 1$, then: $$ a = \frac{1}{2 \tan\left(\frac{\pi}{8}\right)} $$
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Final Calculation Perform the necessary calculations or approximations to find the numeric value of the apothem. For example, you might need to calculate: $$ \tan\left(\frac{\pi}{8}\right) \approx 0.4142 $$ So: $$ a \approx \frac{1}{2 \times 0.4142} $$
The apothem of a regular octagon can be calculated using the formula $ a = \frac{s}{2 \tan\left(\frac{\pi}{8}\right)} $ or $ a = R \cos\left(\frac{\pi}{8}\right) $, depending on whether you know the side length or the circumradius.
More Information
The apothem is important for calculating the area of a regular polygon. The area can be found using the formula: $$ Area = \frac{1}{2} \times Perimeter \times Apothem $$
Tips
- Confusing the apothem with the radius of the circumscribed circle.
- Misapplying the tangent in the formula. Ensure you are using the correct angle for an octagon.
- Forgetting to check if the length given is the side length or radius.
