the arc length and central angle of a circle are 10cm and 80degrees respectivly. what is the length of the radius?

Why did the mathematician take a vacation to the beach? Because he wanted to calculate the sine and cosine of some waves!

But getting back to your question, to find the length of the radius, we need to use a little bit of math. The formula for finding the radius given the arc length (s) and the central angle (θ) is:

𝑟 = 𝑠 / 𝜃

Plugging in the values you provided, we have:

𝑟 = 10cm / 80°

But wait! We need to convert the angle from degrees to radians for this equation. Remember, there are 2π radians in a full circle, which is equivalent to 360 degrees. So, converting 80 degrees to radians:

𝜃 (in radians) = 80° * (2π / 360°)

Now we can substitute this value into the equation:

𝑟 = 10cm / (80° * (2π / 360°))

But why did the circle go to the gym? Because it wanted to get π-thin!

Simplifying the equation and doing the calculations, you'll find the length of the radius.

To find the length of the radius, we can use the formula:

Arc Length = Radius x Central Angle

Given:
Arc Length = 10 cm
Central Angle = 80 degrees

Let's substitute these values into the formula and solve for Radius:

10 cm = Radius x 80 degrees

To find the length of the radius, we need to convert the angle from degrees to radians. There are 2π radians in a full circle, so 360 degrees is equivalent to 2π radians.

To convert 80 degrees to radians, we can use the following formula:

Radian measure = (Degree measure x π) / 180

Radian measure = (80 x π) / 180

Now, let's substitute this value for the central angle in our equation:

10 cm = Radius x [(80 x π) / 180]

Simplifying further:

10 cm = (80π / 180) x Radius

Now, multiply both sides of the equation by (180 / 80π) to isolate the Radius:

(10 cm) x (180 / 80π) = Radius

Simplifying:

Radius ≈ 6.37 cm

Therefore, the length of the radius is approximately 6.37 cm.

To find the length of the radius, we can use the formula relating the arc length, central angle, and radius of a circle. The formula is:

Arc Length = (Central Angle / 360°) × (2πr)

Given that the arc length is 10 cm and the central angle is 80 degrees, we can rearrange the formula to solve for the radius (r):

r = (Arc Length / ((Central Angle / 360°) × 2π)

Plugging in the values, we get:

r = (10 cm / ((80° / 360°) × 2π)

Simplifying further, we have:

r = (10 cm / (0.222 × 2π)

r = (10 cm / 0.138)

r ≈ 72.46 cm

Therefore, the length of the radius is approximately 72.46 cm.

Circumference of a circle

C = 2 r pi

the arc length

L = 2 r pi * theta / 360 °

= r * pi * theta / 180°

10 = r pi * 80 / 180 Multiply both sides by 180

10 * 180 = r * pi * 80

1800 = r * pi * 80 Divide both sides by 80 pi

1800 / 80 pi = r

r = 1800 / 80 pi

r = 40 * 45 / ( 40 * 2 * pi )

r = 45 / 2 pi

r = 45 / ( 2 * 3.14159 )

r = 45 / 6.28318

r = 7.162 cm