Understanding the Theta-Arc-Radius Relationship: Exploring Formulas and Examples

theta = arc /radius

In Mathematics, theta refers to the central angle of a circle, arc refers to a portion of the circumference of a circle, and radius refers to the distance from the center of the circle to any point on its circumference

In Mathematics, theta refers to the central angle of a circle, arc refers to a portion of the circumference of a circle, and radius refers to the distance from the center of the circle to any point on its circumference.

The formula theta = arc / radius relates the central angle (theta) of a circle to the arc length (arc) and the radius. This formula can be used to calculate the central angle when the arc length and radius are known, or it can be used to find the arc length or radius when the central angle is known.

To use this formula, you need to ensure that the arc length and radius are measured in the same units. Let’s go through some examples to better understand how to apply this formula.

Example 1:
If the arc length is 5 cm and the radius is 2 cm, what is the central angle?

Using the formula theta = arc / radius, we can substitute the given values:
theta = 5 cm / 2 cm = 2.5 radians

Therefore, the central angle is 2.5 radians.

Example 2:
If the central angle is 1.2 radians and the radius is 3 m, what is the arc length?

Again, using the formula theta = arc / radius, we can rearrange the formula to solve for arc:
arc = theta * radius

Substituting the given values:
arc = 1.2 radians * 3 m = 3.6 meters

Therefore, the arc length is 3.6 meters.

Example 3:
If the central angle is 60 degrees and the arc length is 8 inches, what is the radius?

To use the formula theta = arc / radius, we need to convert the central angle from degrees to radians since the formula requires radians.

Given that 180 degrees is equal to pi radians, we can set up a proportion to convert:
60 degrees / 180 degrees = theta radians / pi radians

Solving the proportion: 60 / 180 = theta / pi
theta = (60 / 180) * pi = pi/3 radians

Now, using the formula theta = arc / radius and substituting the given values:
pi/3 radians = 8 inches / radius

To solve for the radius, we can cross-multiply and solve for radius:
radius = 8 inches / (pi/3 radians) = (8 inches * 3 radians) / pi

Simplifying further:
radius = 24 inches / pi

Therefore, the radius is 24/pi inches.

Remember to always ensure that your units are consistent when applying this formula.

More Answers:

Understanding the Sine Function: Definition, Evaluation, and Calculation of sin(0)
How to Convert Degrees to Radians: A Step-by-Step Guide with Examples
Converting Radians to Degrees: A Simple Formula for Accurate Angle Conversion

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