Intro to Circles, Central Angles & Arcs, Arc Lengths
The Correct Answer and Explanation is:
Here are the correct answers based on the quiz content:
- Radius: OC
- Diameter: AD
- Chord not a diameter: CB
- Central angle: ∠AOC
- Minor arc: Arc AB
- Major arc: Arc ACB
- Semicircle: Arc ADB
For questions 8 and 9:
- Radius = 18 in Area = π × 18² ≈ 1017.88 in² Circumference = 2π × 18 ≈ 113.10 in
- Radius = 25 m Area = π × 25² ≈ 1963.50 m² Circumference = 2π × 25 ≈ 157.08 m
Question 10:
Given area = 201.06 cm² Solve for radius using A = πr² → r² = 201.06 / π ≈ 63.99 → r ≈ 7.999 Diameter ≈ 16.00 cm Circumference = 2πr ≈ 50.27 cm
Explanation
Understanding circles begins with recognizing their basic parts. A radius is a line segment from the center of the circle to any point on its circumference. The diameter spans the circle through its center, making it twice as long as the radius. A chord is any segment with both endpoints on the circle, and when that chord passes through the center, it becomes the diameter. A central angle is formed when two radii meet at the center and intersect the circle at two points, creating arcs.
Arcs are portions of the circle’s edge. A minor arc is less than half the circle, a major arc is more than half, and a semicircle is exactly half and results from a diameter dividing the circle. Naming arcs involves identifying their endpoints and, if necessary, a third point to clarify their path around the circle.
Calculating area and circumference builds on these basics. The formulas πr² for area and 2πr for circumference allow us to measure the space within and around the circle. Applying these formulas to given radii yields precise numerical results.
In question 10, the area is provided, allowing us to solve for the radius by rearranging the area formula. Taking the square root of the resulting value gives us the radius, which we double for the diameter and then use again to find the circumference.
Mastery of circle geometry requires combining spatial reasoning with formulaic precision. This quiz serves as a comprehensive review of those foundational concepts. If you’d like to dive deeper into arc lengths or sector areas, I’d be glad to walk through them.
