Question 1·Easy·Circles
In the circle shown, radii and form a central angle of . The circle has a circumference of 24.
What is the length of minor arc ?
For arc-length questions with a given central angle and total circumference, quickly set up the proportion . First reduce the angle fraction (like ), then multiply by the circumference; avoid extra steps like finding the radius or using unless the problem specifically requires them.
Hints
Compare the angle to a full circle
A full circle has 360°. How does a 60° angle compare to 360°? Write this as a fraction.
Use the proportion for arc length
Use the idea that . Plug in the central angle and the circumference from the problem.
Finish with multiplication
Once you know what fraction of the circle the 60° angle represents, multiply that fraction by the total circumference, 24, to get the arc length.
Desmos Guide
Compute the fraction of the circumference
In Desmos, type (60/360)*24. The value that Desmos outputs is the length of minor arc .
Step-by-step Explanation
Relate central angle to arc length
For a circle, the length of an arc is proportional to the central angle that intercepts it. This relationship can be written as:
Here, the central angle is and the circumference is 24.
Find what fraction of the circle 60° represents
Compute the fraction of the full circle that a angle represents:
So, minor arc is of the full circumference.
Apply the fraction to the circumference
Use the fraction of the full circumference 24 to find the arc length:
So the length of minor arc is 4, which corresponds to choice B.