Arcs, sectors and reverse angle calculations · Foundation
도입됨| English | 의미 |
|---|---|
| sector/ˈsektə/ | 부채꼴 |
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°.
- This lesson studies sector 扇形: A circle region bounded by two radii and their intervening arc.
The method
- 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.
A sector has r=6 cm and arc 4π cm. Its perimeter is 4π+12, not 4π. Why?
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
A sector is bounded by two radii AND the arc. The arc 4π cm alone omits the two straight edges of 6 cm each; the complete perimeter is 4π+12 cm.
A worked case
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.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
(120/360)×2×6=4.
Where it goes wrong
- 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.
Not true: A sector’s arc length and perimeter are the same. The perimeter also includes the two radii: perimeter = arc + 2r.
Find the coefficient of pi in its area.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
(120/360)×6²=12.
Sector means a circle region bounded by two radii and their intervening arc.
난이도 1/5 · 100 코인 + 오늘 첫 완료 시 소액 보너스
True: A circle region bounded by two radii and their intervening arc.
Applying it
- 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.
Find θ for r=6 sector area 9π.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
9/36 of a turn: 360×1/4=90°.
Match each sector quantity for angle θ° and radius r to its expression.
난이도 2/5 · 140 코인 + 오늘 첫 완료 시 소액 보너스
Each is the fraction θ/360 of the corresponding whole-circle quantity; only the area squares the radius, and only the perimeter adds the radii.
In your course
- AQA GCSE Mathematics (8300), Foundation tier, specification section 3.4.
- Paper 1 is non-calculator: keep fractions, surds and π exact and show each step. On calculator papers, still write the expression before its value.
- Sector: A circle region bounded by two radii and their intervening arc.
- For an angle θ in degrees, fraction of a turn is θ/360.
- Watch for: an arc length is not a sector perimeter.