How to Calculate Arc Length Using Radians

TL;DR

To find the length of an arc with a central angle of 3/2 radians and a radius of 6 centimeters, use the formula s = rθ. Here, s is the arc length, r is the radius, and θ is the angle in radians. Plugging in the numbers gives an arc length of 9 centimeters.

Transcript

In the circle below, central angle GOH has a measure of 3/2 radians. What is the length of arc GH? So let's define some variables that will help us manipulate them a little bit easier. So where is central angle GOH? It's this angle right over here. Let's call that theta. So let's say that theta is equal to the measure of angle GOH. Now they're aski... Read More

Key Insights

• 🫠 Radians measure angles in terms of arc lengths, providing a more direct relationship between angles and arcs.
• 🫠 The definition of a radian states that an angle in radians is the ratio between the length of the arc it subtends and the radius of the circle.
• 🫠 The equation s = rθ allows us to find the length of an arc given the central angle and the radius of the circle.

Questions & Answers

Q: How does an angle measured in radians relate to the length of the arc subtended by that angle?

An angle measured in radians is the ratio between the length of the arc it subtends and the radius of the circle. This relationship is defined by the equation s = rθ, where s is the arc length, r is the radius, and θ is the angle in radians.

Q: How is the equation s = rθ used to find the length of arc GH?

In this problem, the central angle GOH is given as 3/2 radians, and the radius is 6 centimeters. Plugging these values into the equation, we get s = 6 cm * 3/2 radians. Simplifying, the length of arc GH is calculated to be 9 centimeters.

Q: Can the equation s = rθ be used for any circle, regardless of the radius?

Yes, the equation s = rθ holds true for any circle, whether it is a unit circle or not. The ratio between the arc length and the angle in radians remains the same, regardless of the circle's size.

Q: Is the length of arc GH dependent on the measure of the central angle?

Yes, the length of arc GH is directly proportional to the measure of the central angle. As the angle increases, the length of the arc also increases, and vice versa, as long as the radius remains constant.

Summary & Key Takeaways

• The video discusses how to find the length of an arc in a circle using radians.

• The central angle GOH is given as 3/2 radians, and the radius of the circle is 6 centimeters.

• By using the definition of radian measure, the equation s = rθ is derived, where s represents the length of the arc, r is the radius, and θ is the angle in radians.

• By plugging in the values, the length of the arc GH is calculated to be 9 centimeters.