Arcs, sectors and reverse angle calculations · Higher
| English | 中文 | Pinyin |
|---|---|---|
| sector/ˈsektə/ | 扇形 | shàn xíng |
A circular fan opens through 120°. It covers one third of a full turn, so its arc and area are each one third of the corresponding whole circle.
- A circular fan opens through 120°. It covers one third of a full turn, so its arc and area are each one third of the corresponding whole circle.
- This lesson studies sector 扇形: A circle region bounded by two radii and their intervening arc.
Choose the mathematical structure
- For an angle θ in degrees, fraction of a turn is θ/360. Arc length is this fraction of 2πr and sector area is this fraction of πr². Sector perimeter adds the two radii. Rearrange the same fraction to recover an angle from an arc or an area. Keep degree and length units separate.
- State the allowed inputs and units before calculating. An equation should express the relationship, not just record a calculator entry.
Which description correctly defines sector?
A circle region bounded by two radii and their intervening arc.
Work through a checked case
- Check the result against the starting quantities. Substitute into the original relation, or compare the graph and numerical answer where appropriate.
At r=6 cm and θ=120°, the fraction is 1/3: arc is 4π cm, area is 12π cm² and perimeter is 4π+12 cm. If another r=6 sector has area 9π cm², its fraction is 9π/36π=1/4 and angle is 90°. If its arc instead measures 3π cm, its fraction is 3π/12π=1/4, giving the same angle. A full 360° sector has the whole circle area, while the circle boundary has no extra radii. A 60° sector of radius 3 has area (1/6)×9π=1.5π cm².
Arcs, sectors and reverse angle calculations
For an angle θ in degrees, fraction of a turn is θ/360
Classify the worked-case statements, then explain the units or invariant that justifies each decision.
For r=6,θ=120°, find the coefficient of pi in arc length.
(120/360)×2×6=4.
Test a tempting shortcut
- An arc length is not a sector perimeter. Use the angle as a fraction of 360°, not 180°. Radius is squared only for area, not arc length.
- When a shortcut fails, identify the assumption it breaks. Keep an exact value until the requested final rounding.
A sector’s arc length and perimeter are always the same. This claim is false. Explain which definition or assumption it violates.
Find the coefficient of pi in its area.
(120/360)×6²=12.
A sector’s arc length and perimeter are always the same.
An arc length is not a sector perimeter. Use the angle as a fraction of 360°, not 180°. Radius is squared only for area, not arc length.
Interpret a new situation
- AQA G18 includes arc lengths, sector angles and areas in degree-based geometry. Show the full-circle quantity and fraction before multiplying, then check that the result fits the angle.
- A complete solution gives the mathematical result and explains what it means. Check that it is possible in the stated context.
Find θ for r=6 sector area 9π.
9/36 of a turn: 360×1/4=90°.
Match each part of a complete solution to its purpose.
An assumption justifies the model; a check tests the result; interpretation connects it to the question.
Use this in your course
- 8300 · Higher · 3.4. Match the target tier and specification before assigning extensions.
- Give the method before the final answer, and use the paper's calculator and formula rules. Review a wrong answer by locating the first invalid step.
A circle region bounded by two radii and their intervening arc. Choose the relationship, show the method, check its assumptions and interpret the result.