Area of a triangle in terms of the tangent function

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Discussion Overview

The discussion revolves around the area of a triangle formed by one side of a circumscribed n-gon and the lines connecting the endpoints of that side to the center of the inscribed circle. Participants explore the relationship between the triangle's area and the tangent function, particularly focusing on the expression tan(π/n).

Discussion Character

  • Exploratory
  • Mathematical reasoning
  • Debate/contested

Main Points Raised

  • Some participants propose that the area of the triangle is given by tan(π/n), but question the validity of this for n=2, suggesting that it leads to an infinite area.
  • Others argue that the base and height of the triangle can be computed as b = 2 sin(π/n) and h = cos(π/n), leading to an area of A = sin(π/n) cos(π/n).
  • A participant notes that the radius of the circle is necessary for determining the area, suggesting that the area might be r² tan(π/n) if the radius is not specified.
  • Another participant inquires about the angle of the sector corresponding to the triangle and suggests drawing a picture to clarify the relationships between the sides.
  • One participant attempts to apply the Pythagorean theorem to find side lengths but is uncertain if this is the correct approach, indicating a need for clearer definitions of the triangle's dimensions.
  • Another participant suggests simplifying the notation for side lengths to facilitate calculations and encourages focusing on the right triangle formed by the apothem and the sides.

Areas of Agreement / Disagreement

Participants express differing views on the correct approach to calculating the area of the triangle, with no consensus reached on the validity of the initial claim regarding tan(π/n) or the subsequent calculations involving sine and cosine functions.

Contextual Notes

Participants highlight the importance of the circle's radius and the definitions of side lengths, indicating that assumptions about these parameters significantly affect the calculations and conclusions drawn.

Bushy
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Hi there, the problem says, an n-gon is circumscribed around a circle so the mid point of each side is tangent to the circle.

Prove the triangle consisting of one side of the n-gon and the sides from the end points to the middle of the circle has area

tan(pi/n)

Cheers!
 
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Re: Area of a triangle in terms of tan

Bushy said:
Hi there, the problem says, an n-gon is circumscribed around a circle so the mid point of each side is tangent to the circle.

Prove the triangle consisting of one side of the n-gon and the sides from the end points to the middle of the circle has area

tan(pi/n)

Cheers!

... so that for n=2 is $\displaystyle A= \tan \frac{\pi}{2} = \infty$?... it seems a little improbable so that let's compute the base and the height of each triangle...

$\displaystyle b = 2\ \sin \frac{\pi}{n}$ $\displaystyle h = \cos \frac{\pi}{n}$ ... and the area is... $\displaystyle A= \frac{b\ h}{2} = \sin \frac{\pi}{n}\ \cos \frac{\pi}{n}$ Kind regards $\chi$ $\sigma$

P.S. I mistakenly undestood 'inscribed' instead of 'circumscribed' and that explains my answer... very sorry! (Wasntme)...
 
Last edited:
Re: Area of a triangle in terms of tan

Bushy said:
Hi there, the problem says, an n-gon is circumscribed around a circle so the mid point of each side is tangent to the circle.

Prove the triangle consisting of one side of the n-gon and the sides from the end points to the middle of the circle has area

tan(pi/n)

Cheers!

Does the problem mention the radius of the circle? The area of the triangle you describe is going to have to depend on $r$. I could believe the result either if you were asked to show that the area is $r^{2} \tan \left( \tfrac{ \pi}{n} \right)$, or if the circle was assumed to have a radius of $1$.
 
Re: Area of a triangle in terms of tan

Yes the radius is one. How did you get tan from there?
 
Re: Area of a triangle in terms of tan

So, imagine a sector of the circle, corresponding to the triangle whose area you want to find. What is its angle? Start labeling sides, and then form the area of the triangle. What does that give you? (I'd recommend drawing a picture!)
 
Re: Area of a triangle in terms of tan

Ackbach said:
So, imagine a sector of the circle, corresponding to the triangle whose area you want to find. What is its angle? Start labeling sides, and then form the area of the triangle. What does that give you? (I'd recommend drawing a picture!)

The angle is 2*pi /n, I cannot find any side lengths.

If I halve the trianglefrom the centre of the circle the angle becomes pi/n and one of the side lengths becomes 1 from the radius. If we call the lengthof the tangent = AB then the other side length from the circle becomes

(1^2+(1/2 AB)^2)^(1/2) using the right angle.

Not sure if I am heading off track here...
 
Re: Area of a triangle in terms of tan

Bushy said:
The angle is 2*pi /n, I cannot find any side lengths.

If I halve the trianglefrom the centre of the circle the angle becomes pi/n and one of the side lengths becomes 1 from the radius. If we call the lengthof the tangent = AB then the other side length from the circle becomes

(1^2+(1/2 AB)^2)^(1/2) using the right angle.

Not sure if I am heading off track here...

You've definitely made some progress, but I don't think the Pythagorean Theorem is the best thing to do next. You don't, off-hand, know the length of the side from the point of tangency to the edge of the polygon side. So call it something. Do you now have a triangle you can work with?
 
Re: Area of a triangle in terms of tan

Ackbach said:
You've definitely made some progress, but I don't think the Pythagorean Theorem is the best thing to do next. You don't, off-hand, know the length of the side from the point of tangency to the edge of the polygon side. So call it something. Do you now have a triangle you can work with?
That length was not given, I have called it 1/2*AB in my attempt above.
 
Re: Area of a triangle in terms of tan

Bushy said:
That length was not given, I have called it 1/2*AB in my attempt above.

That seems an overly complicated name for it. I would just go with something like $y$. So you have a right triangle. The angle at the center of the polygon is $\pi/n$. The apothem is $1$. From that data, can you compute $y$? And then what is the area of the right triangle? And then what is the area of the whole triangle?
 

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