Area of a Circle: Solving the Equation

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In summary: A}{4} = r \int_0^{\pi/2} \cos^2 t \mathrm{d} t = r \int_0^{\pi/2} \dfrac{1 + \cos 2t}{2} \mathrm{d} t = \dfrac{r \pi}{4} #### \displaystyle A = r^2 \pi ##You know that the area of a circle is ##\pi r^2##, so what's the problem?
  • #1
Vector1962
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TL;DR Summary
area of a circle in terms of Y if center of circle is at (0 , r) --> A=f(y)
i can write the equation of circle easy enough, x^2+(y-r)^2=r^2. i get A=r^2/2 * asin((y-r)/r) + (y-r)/2 * sqrt(r^2 - (y-r)^2) through integration (using change of variable). Letting u = (y-r) and u^2=(y-r)^2, du= dy. Here's the rub... it's not right... :-) Appreciate and thanks in advance for any pointers... it's been a long time since I've done anything like this.
 
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  • #2
Vector1962 said:
Summary: area of a circle in terms of Y if center of circle is at (0 , r) --> A=f(y)

i can write the equation of circle easy enough, x^2+(y-r)^2=r^2. i get A=r^2/2 * asin((y-r)/r) + (y-r)/2 * sqrt(r^2 - (y-r)^2) through integration (using change of variable). Letting u = (y-r) and u^2=(y-r)^2, du= dy. Here's the rub... it's not right... :-) Appreciate and thanks in advance for any pointers... it's been a long time since I've done anything like this.
I'm not sure I know what you are doing. Normally, you would calculate the area of half or a quarter of the circle using ##x^2 + (y - r)^2 = r^2##. If you try to do the whole circle, then ##y## is not a single-valued function of ##x##.
 
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  • #3
Vector1962 said:
i get A=r^2/2 * asin((y-r)/r) + (y-r)/2 * sqrt(r^2 - (y-r)^2) through integration (using change of variable).
I agree with @PeroK's comment. Seeing only your result, but not the integral you used, it's hard to say why your result is wrong.
 
  • #4
Here is a start for you

## x^2 + (y-r)^2 = r^2 ##

##u = y-r##

## \displaystyle \dfrac{A}{4} = \int_0^r \sqrt{r^2 - x^2}\mathrm{d} x
= r \int_0^r \sqrt{1 - x^2/r^2}\mathrm{d} x##

##x= r \sin t##, ##x = 0 ##gives ##t = 0##, ##x = r## gives ##t = \pi / 2 ##

##\mathrm{d}x = r \cos t \mathrm{d}t ##

## \displaystyle \dfrac{A}{4} = r\int_0^{\pi/2} \sqrt{1 - \sin^2t} \cos t \mathrm{d} t
##
 

Related to Area of a Circle: Solving the Equation

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

The formula for finding the area of a circle is A = πr², where A is the area and r is the radius of the circle.

2. How do you solve for the area of a circle?

To solve for the area of a circle, plug the given radius value into the formula A = πr² and calculate the result using the value of π (pi) as 3.14 or a more accurate value if needed.

3. Can you find the area of a circle without knowing the radius?

No, the radius is a necessary component in the formula for finding the area of a circle. If the radius is not given, it cannot be solved for.

4. How do you convert the diameter to the radius in the area of a circle formula?

To convert the diameter to the radius, divide the diameter by 2. The resulting value will be the radius, which can then be used in the formula A = πr² to find the area of the circle.

5. Is there a shortcut or easier way to find the area of a circle?

Yes, there is a shortcut formula for finding the area of a circle when the diameter is known. The formula is A = (π/4) x D², where A is the area and D is the diameter of the circle.

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