The circle has a radius of 6. An arc in this circle has a central angle of 48 degrees. What is the length of the arc? Enter an exact answer in terms of π or use 3.14 for π and enter your answer as a decimal.
SAT · High School · Sun Jan 24 2021
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he length of an arc (L) in a circle is determined by the formula: \[ L = \frac{\theta}{360} \times 2\pi r \] where: - \( \theta \) is the central angle in degrees, - \( r \) is the radius of the circle, - \( \pi \) is a mathematical constant, approximately equal to 3.14159.
Given that the radius of the circle (r) is 6 and the central angle (θ) is 48 degrees, plugging these values into the formula yields:
\[ L = \frac{48}{360} \times 2\pi \times 6 \]
To simplify: \[ L = \frac{48}{360} \times 2\pi \times 6 \] \[ L = \frac{48}{60} \times \pi \times 6 \] \[ L = \frac{4}{5} \times \pi \times 6 \] \[ L = \frac{24}{5} \pi \] \[ L = 4.8\pi \]
If we use \( \pi \approx 3.14 \) to find a decimal approximation: \[ L = 4.8 \times 3.14 \] \[ L \approx 15.072 \]
So the length of the arc is \( 4.8\pi \) (exact in terms of π) or approximately 15.072 units when \( \pi \) is taken as 3.14.
Extra: Understanding the concept of an arc in a circle is important in geometry. An arc is a segment of the circumference of a circle. The circumference is the total distance around the circle, and it can be calculated using the formula \( C = 2\pi r \), where \( r \) is the radius of the circle. The length of an arc, therefore, is a portion of the circumference.
The central angle of the arc, often measured in degrees, indicates what fraction of the entire circle's circumference the arc represents. Since there are 360 degrees in a full circle, if you know the central angle, you can simply divide that angle by 360 to find the fraction of the whole circumference that the arc length represents. Multiplying this fraction by the full circumference \( (2\pi r) \) gives you the arc length. This is very useful in various problems in geometry, such as finding the length of sectors, which are like slices of a circular "pie", or in practical situations like determining the distance along a circular track or wheel.