How to Find Arc Length and Sector Area? (+FREE Worksheet!)
Arc length is the distance along part of a circle, and sector area is the slice of the circle enclosed by that arc. Both are fractions of the whole: multiply the circumference or the area by the central angle over 360 degrees.
Arc Length and Sector Area, Example 4:
Exercises for Arc Length and Sector Area
Arc Length and Sector Area.
1.(color{blue}{r=10 ft,θ=25^°})
Tutor-style math help
Find Arc Length and Sector Area: what to notice and how to work it
Trigonometry skill
Trigonometry connects an angle to a triangle ratio, a unit-circle coordinate, or a repeating graph. Choosing the right picture makes the problem much easier.
What to notice first
Decide whether the problem is triangle-based, circle-based, or graph-based. Then use the matching definition.
Common student mistake
Do not mix degrees and radians. The angle unit must match the formula, graph scale, or calculator setting.
Key formulas and cues
(sintheta=frac{text{opposite}}{text{hypotenuse}})
(costheta=frac{text{adjacent}}{text{hypotenuse}})
(tantheta=frac{sintheta}{costheta})
(sin^2theta+cos^2theta=1)
A reliable path
- Choose the modelUse a right triangle, the unit circle, or a transformed graph.
- Track unitsConvert degrees and radians when needed.
- Use identitiesReplace complicated trig expressions with equivalent simpler ones.
Worked examples
Right-triangle sine
Example: opposite = 5, hypotenuse = 13
- Sine is opposite over hypotenuse.
- Substitute 5 and 13.
- Leave the ratio simplified.
Answer: (sintheta=frac5{13})
Unit-circle cosine
Example: (cos(0))
- At angle 0, the point is (1, 0).
- Cosine is the x-coordinate.
- Read the x-value.
Answer: (1)
Try one before moving on
Try: In a right triangle, tangent equals which ratio?
Answer: Opposite over adjacent.
Next step: do the matching worksheet or quiz while the method is still fresh, then come back and explain the first step in your own words.
x
Find Arc Length and Sector Area: pop-up practice
Answer these quick questions, then use the feedback to decide which part of the lesson to review.
Choose an answer to begin.
1. Cosine in a right triangle is:
2. (180^circ) equals:
3. The unit circle has radius:
2.(color{blue}{r=12 cm,θ=30^°})
3.(color{blue}{r=9 cm,θ=27^°})
4.(color{blue}{r=11 ft,θ=44^°})
Find the area of each sector. ((π=3.14))
5.(color{blue}{r=5 ft,θ=40^°})
6.(color{blue}{r=6 cm,θ=60^°})
Length of the arc:
1.(color{blue}{4.36 ft})
2.(color{blue}{6.28 cm})
3.(color{blue}{4.24 cm})
4.(color{blue}{8.44 ft})
Area of a sector:
5.(color{blue}{8.72 ft^2})
6.(color{blue}{18.84 cm^2})
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