Arc Length of a Circle: Learn the Proof!

• Miike012
In summary, the conversation discussed finding the arc length of a circle with a given radian and radius. The formula for finding the arc length is s = radian * radius, but it only works for radians and not degrees. To convert from degrees to radians, the formula is s = (pi/180) * theta * radius. The correct formula was corrected by one of the participants in the conversation.
Miike012

Homework Statement

Today we went over finding the arc length s of a circle with a given radian and radius...

Thats easy to remember but I think it will be more memorable for the long run if I knew the proof and understood it... can some one please post a website where I can read the proof... or if some one could explain that would be nice to .
Thank you.

The Attempt at a Solution

The arc length of the entire circle (its circumference) of radius r is $2 \pi r$. IOW, the arc length associated with an angle of $2 \pi$ is $2 \pi r$. Since the arc length is proportional to the angle between the two rays that subtend the arc, the arc length associated with an angle $\theta$ is $\theta r$.

But this formula only works if I am presented with radians correct? So if I am given deg. I will have to convert into rad right?

Miike012 said:
But this formula only works if I am presented with radians correct? So if I am given deg. I will have to convert into rad right?

That's correct. Alternately, you could use this formula:

$$s=\frac{180}{\pi} \theta r$$

Where $\theta$ is measured in degrees.

Say 64 is the deg. w/ radius of 1
Then your saying I can multiply 180/pi*64*1
= 3666... that seems to big to be an arc length of radius 1

That's because the formula is wrong. It should be
$$s=\frac{\pi}{180} \theta r$$

Thank you.

eumyang said:
That's because the formula is wrong. It should be
$$s=\frac{\pi}{180} \theta r$$

Oh, my bad... I got the conversion wrong, I guess.

Sorry, Miike!

Char. Limit said:
Oh, my bad... I got the conversion wrong, I guess.

Sorry, Miike!
Its cool no big deal

1. What is the arc length of a circle?

The arc length of a circle is the distance along the circumference of the circle between two points on the circle, measured in linear units such as centimeters or inches.

2. How is the arc length of a circle calculated?

The arc length of a circle can be calculated using the formula s = rθ, where s is the arc length, r is the radius of the circle, and θ is the central angle in radians.

3. Why is the arc length of a circle important?

The arc length of a circle is important because it helps in determining the distance traveled by an object moving along the circumference of a circle, such as the hands of a clock or a rotating wheel. It is also used in various real-world applications such as navigation and engineering.

4. Can the arc length of a circle be greater than the circumference?

No, the arc length of a circle cannot be greater than the circumference. The circumference is the distance around the entire circle, while the arc length is the distance between two points on the circle. Therefore, the circumference is always larger than the arc length.

5. How is the proof for the arc length of a circle derived?

The proof for the arc length of a circle can be derived using calculus, specifically by finding the limit of a sum of infinitely small line segments along the circumference of the circle. This leads to the formula s = rθ and shows the relationship between the arc length and the central angle in radians.

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