Area of a triangle in terms of the tangent function

In summary: Re: Area of a triangle in terms of tanThat 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?y=\frac{1}{2} \sin \frac{\pi}{n}Area of the triangle=\sin(\frac{\pi}{n})Area of the whole triangle=\sin(\frac{\pi}{n}+\frac
  • #1
Bushy
40
0
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!
 
Mathematics news on Phys.org
  • #2
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:
  • #3
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 $1$.
 
  • #4
Re: Area of a triangle in terms of tan

Yes the radius is one. How did you get tan from there?
 
  • #5
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!)
 
  • #6
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...
 
  • #7
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?
 
  • #8
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.
 
  • #9
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?
 

Related to Area of a triangle in terms of the tangent function

What is the formula for finding the area of a triangle using the tangent function?

The formula for finding the area of a triangle in terms of the tangent function is A = (1/2)bh, where b is the length of the base and h is the height of the triangle. This formula applies to right triangles, where the base and height are the sides adjacent and opposite to the right angle, respectively.

Can the tangent function be used to find the area of any triangle?

No, the tangent function can only be used to find the area of right triangles. For other types of triangles, different formulas such as the Heron's formula must be used.

How is the tangent function related to the area of a triangle?

The tangent function is related to the area of a triangle through the trigonometric ratios of the sides of the triangle. In a right triangle, the tangent of an acute angle is equal to the ratio of the opposite side to the adjacent side. This relationship is used in the formula for finding the area of a right triangle in terms of the tangent function.

Can the area of a triangle be negative when using the tangent function?

No, the area of a triangle can never be negative. The tangent function may produce negative values, but the area is always calculated as a positive value.

How accurate is the area of a triangle calculated using the tangent function?

The area of a triangle calculated using the tangent function can be accurate as long as the measurements of the base and height are accurate. However, due to the rounding of decimals and the use of approximations in the calculation of the tangent function, there may be a slight margin of error in the final result.

Similar threads

Replies
1
Views
848
Replies
2
Views
936
  • General Math
Replies
2
Views
1K
  • New Member Introductions
Replies
1
Views
130
Replies
1
Views
1K
Replies
2
Views
1K
Replies
30
Views
4K
  • General Math
Replies
9
Views
2K
  • General Math
Replies
1
Views
1K
Replies
1
Views
2K
Back
Top