Understanding the Angle Subtended at the Center of a Circle by an Arc of a Given Length

When dealing with circles, it is often necessary to determine the angle subtended at the center of the circle by a given arc length. This article will guide you through the steps to calculate this angle in degrees.

The Given Parameters

In this specific scenario, we have a circle with a radius of 3 units and an arc length of 1 unit. The relationship between the arc length s, the radius r, and the central angle in radians θ is defined by the formula:

s rθ

Calculating the Central Angle in Radians

Given the values, we can rearrange the formula to solve for the angle θ in radians:

θ s/r
θ 1/3 radians

Converting Radians to Degrees

To convert the angle from radians to degrees, we use the conversion factor:

degrees radians × (180/π)

Substituting the value of θ in radians:

θ in degrees (1/3) × (180/π)
θ ≈ 19.1 degrees

Summary of the Calculation

Given a circle with a radius of 3 units and an arc length of 1 unit, the angle subtended at the center of the circle is approximately 19.1 degrees. This can be calculated using the relationship between the arc length, the radius, and the central angle, and then converting the angle from radians to degrees.

Other Relevant Calculations

There are a few other methods to calculate the angle, as shown below:

Circumference of the Circle

The circumference of the circle is given by:

C 2πr 2 × 22/7 × 3 132/7 cm
This represents 360 degrees. If the arc length is 1 cm, the angle subtended is:

(360 × 7/132) ≈ 19.09 degrees

Radical Calculation

The angle in radians is 1/3. To convert radians to degrees, use the formula:

1/3 × (180/π) ≈ 19.096°

Definition-Based Calculation

By definition, the angle subtended in radians is the arc length divided by the radius. Therefore, for 1/3 radians, the angle converted to degrees is:

(60/π) ≈ 19.1 degrees

Conclusion

In summary, to find the angle subtended at the center of a circle by a given arc length, you can use the formula s/r to find the central angle in radians, and then convert it to degrees using the conversion factor 180/π. This method, along with other related calculations, will help you understand and solve similar problems involving circles, arcs, and angles.