Question

A circle with radius r and arc s intercepted by central angle theta is shown.





Question
Explain why the arc length, s, divided by the radius is the same, regardless of the circle's radius. Move words to the lines and expressions to the boxes to complete the statements.

Response area with 7 blank spaces
There are
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radians in a circle, so the ratio of the
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to the entire circle is given by the expression
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. The circumference of a circle with radius r is given by the expression
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. The arc length, s, is given by the expression
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. Dividing the expression for arc length by r gives the expression
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. Thus, the arc length divided by the radius is always equal to the measure of the
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, regardless of the radius.
Answer options with 11 options.
Answer Options
2 pi
pi r squared
2 pi r
theta
StartFraction theta over 2 pi EndFraction

Answer 1

There are 2π radians in a circle, so the ratio of the theta to the entire circle is given by the expression theta/(2π). The circumference of a circle with radius r is 2πr. The arc length s is (theta/(2π))·(2πr) = r·theta. Dividing the expression for arc length by r gives theta. Thus, the arc length divided by the radius is always equal to the measure of the central angle (theta), regardless of the radius.

Filled blanks:
- Blank 2: 2π
- Blank 3: theta
- Blank 4: theta/(2π)
- Blank 5: 2πr
- Blank 6: r·theta
- Blank 7: theta
- Blank 8: central angle (theta)