Area, approximating triangles?

In summary, the author is trying to show that the area of an n sided inscribed polygon is 0, but his equation is wrong.
  • #1
rocomath
1,755
1
Let [tex]A_n[/tex] be the area of a polygon with n equal sides inscribed in a circle with radius r. By dividing the polygon into n congruent triangles with central angle [tex]\frac{2\pi}{n}[/tex], show that [tex]A_n=\frac 1 2 \pi r^2\sin{\frac{2\pi}{n}}.[/tex]

Ok, I drew a circle with congruent triangles inscribed in it. I assumed that it was an equilateral triangle, so it has height [tex]\frac{\sqrt{3}}{2}r[/tex].

So far I have

[tex]A_{triangle}=\frac 1 2 \cdot r \cdot \frac{\sqrt{3}}{2}r[/tex]

[tex]\sin{\frac{2\pi}{n}}=\frac{\frac{\sqrt{3}}{2}r}{r}=\frac{\sqrt{3}}{2}[/tex]

[tex]A_{triangle}=\frac 1 2 \cdot r^2\cdot \sin{\frac{2\pi}{n}}[/tex]

Now I'm stuck, maybe my assumption was incorrect, and I also do not know how to incorporate the fact that it is inscribed in the circle. I know I need to take it into consideration noticing that it wants me to express the answer with the area of a circle as part of the answer. Or perhaps [tex]\pi r^2[/tex] appears through substitutions?
 
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  • #2
I remember learning a variation of this years ago in high school, this is a nifty little formula

The first thing I see is that I'm not sure you even understood what you were being asked to prove

Do it with a simple shape, like a hexagon(I tried an octagon myself but couldn't draw a circle worth a darn that circumscribed it :( )

Draw the circle around it that touches every intersection on the hexagon. Now from the center of the circle, draw a line to every intersection and behold six triangles!

Note that they won't necessarily be equilateral triangles since two sides are the radius of the circle and one's a chord(I think that's the term >_>) Isosceles always though, I think

So what's the area of that triangle? The base is r, you need 1/2*base*height, the height you have to drop a perpendicular and find that, you need the sine of that angle...well you have the full circle broken into 6 things, so...

Anyways that's a better way to start
 
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  • #3
1) I don't know what an equilateral triangle has to do with anything if you have n sides. 2) Your A_n approaches 0 as n approaches infinity, hence A_n is NOT the area of a polygon with n equal sides inscribed in a circle of radius r. Look, what's the area of an isosceles triangle with apex angle 2pi/n? Multiply that by n to get the total area.
 
  • #4
blochwave said:
Note that they won't necessarily be equilateral triangles since two sides are the radius of the circle and one's a chord(I think that's the term >_>) Isosceles always though, I think
LOL, I assumed the chord was length r, hence the equilateral triangle. Ok, let me continue reading you and Dick's post. Must solve this!
 
  • #5
Right, which is basically where I went

I googled, and as feared, his equation is wrong, I think you misread pi for n
 
  • #6
blochwave said:
Right, which is basically where I went

I googled, and as feared, his equation is wrong, I think you misread pi for n
Confused pi for n? That is the final equation it wants though.

[tex]A_n=\frac 1 2 \pi r^2\sin{\frac{2\pi}{n}}[/tex]

Stewart 5th edition, page 326
 
  • #7
rocophysics said:
Confused pi for n? That is the final equation it wants though.

[tex]A_n=\frac 1 2 \pi r^2\sin{\frac{2\pi}{n}}[/tex]

Stewart 5th edition, page 326

If that's supposed to be the area of an n sided inscribed polygon, it's wrong. There must be a typo in "Stewart 5th edition, page 326".
 
  • #8
My calc 3 professor was a proofreader for math textbook solutions

I doubt he was very good >_>

Of course every time I've been so certain I'm right and the book's wrong I've been just ludicrously wrong, still it's not too surprising.
 

Related to Area, approximating triangles?

1. What is the formula for finding the area of a triangle?

The formula for finding the area of a triangle is (base x height)/2. This means that you multiply the base of the triangle by the height and then divide the answer by 2.

2. How do you approximate the area of an irregular triangle?

To approximate the area of an irregular triangle, you can divide it into smaller, regular triangles. Then, use the formula (base x height)/2 for each smaller triangle and add the results together.

3. Can you use the Pythagorean theorem to find the area of a triangle?

No, the Pythagorean theorem is used to find the length of the sides of a right triangle, not the area.

4. Does the order in which you measure the base and height of a triangle matter?

No, the order in which you measure the base and height of a triangle does not matter. As long as you use the correct values in the formula (base x height)/2, you will get the same area.

5. What are some real-world applications of approximating the area of triangles?

Approximating the area of triangles is used in many fields, such as architecture, engineering, and cartography. It can also be applied in everyday tasks, such as calculating the area of a piece of land or the surface area of a roof.

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