Angles (III): sector angles

1Summary 重點摘要

Sector angle = share × 360°.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: A full turn is 360°The angles around a circle total 360°: when a clock hand goes once around, it sweeps exactly 360°. Why can no sector calculation do without 360°? Because a sector is a part of the circle, and what fraction of the 360° its central angle takes is precisely the basis of every calculation; the sectors of one circle together can never have angles beyond that total. For example, cutting a pizza into 6 equal slices gives each slice a central angle of 360 ÷ 6 = 60°, and 6 slices make exactly 360°. People chose 360 for the full turn because it divides exactly by 2, 3, 4, 5, 6 and many other numbers, which makes splitting especially convenient. A common mistake is to think a half circle is 360°; a half circle is half of the whole, 180°, and only the full turn is 360°.
2. Step 2: The formulaSector angle = share of the circle × 360°. For a sector of 14, multiply 14 by 360°. Why multiply? Because 'a fraction of the circle' means taking a part of the whole, and finding a fraction of a number uses multiplication: a quarter of 360° is 360 × 14. The fixed plan for these questions is to find the share first and then multiply by 360°: a sector of 16 has central angle 360 × 16 = 60°, and a sector of 34 has 360 × 34 = 270°. A common mistake is to turn the fraction upside down and divide 360 by the numerator, or to skip simplifying so the arithmetic grows heavy. Divide before multiplying: for 360 × 34, work out 360 ÷ 4 = 90 first and then multiply by 3 to get 270°, quick and accurate.
3. Step 3: The quarter caseA sector of 14: 360 × 14 = 360 ÷ 4 = 90°, exactly a right angle. Why does a quarter circle match a right angle? Because a right angle is 90°, and four such sectors put together give 90 × 4 = 360°, covering the circle exactly. A quarter-circle sector is also called a right-angle sector: a pizza cut into four equal slices has a 90° angle at the tip of every slice, and the sector of a clock face from 12 to 3 is likewise 90°. The example also offers a shortcut: seeing 'a quarter' gives 90° at once, and seeing 'a half' gives 180°; fix these two anchors first and work other fractions out from them. A common mistake is to compute 360 × 14 as 360 ÷ 1 × 4 = 1440°; when a fraction multiplies a whole number, the denominator divides and the numerator multiplies, and that order may not be swapped.
4. Step 4: Percentages work the sameA sector given as a percentage of the circle is computed by the same multiplication: a sector of 25% has central angle 360 × 25% = 90°; a sector of 50% has 360 × 50% = 180°; a sector of 10% has 360 × 10% = 36°. Why is the method identical to fractions? Because a percentage is a fraction with denominator 100 to begin with — 25% is 25100, that is 14; before multiplying, change the percentage into a fraction or a decimal, for example 25% into 0.25, giving 360 × 0.25 = 90°. The pie chart in statistics is exactly this idea in action: each sector stands for the percentage of a part, and the bigger the share, the bigger the angle. A common mistake is to leave the percent sign in and take 360 × 25 as 9000°; convert to a decimal or a fraction before multiplying — the % sign never enters the calculation.
5. Step 5: From angle back to shareThe reverse also works: from the central angle, find what fraction of the circle the sector is. A 120° sector gives 120 ÷ 360 = 13, so it takes one third of the circle; a 45° sector gives 45 ÷ 360 = 18. Why divide? Because finding 'what fraction a part is of the whole' uses the part divided by the whole — the sector's angle divided by the circle's 360°. Multiplying and dividing undo each other: 13 of 360 returns to 120°, and working in both directions keeps every sector question flexible; knowing a central angle of 90°, for instance, tells you the share is 14, that is 25%. A common mistake is to compute 360 ÷ 120 = 3 and offer 3 as the answer; 360 ÷ 120 = 3 says the whole is 3 times the sector, so the sector takes 13 — after dividing, always be clear about who is divided by whom and who takes what share.
6. Step 6: Clock applicationA clock face is a circle split into 12 large marks, each mark 360 ÷ 12 = 30°. At 3 o'clock the hour hand points at 3 and the minute hand at 12; the hands are 3 marks apart, so the angle between them is 3 × 30° = 90°. At 6 o'clock they are 6 marks apart, the angle is 180°, and the hands form a straight line. Why use 30° per mark? Because the face divides the full 360° into 12 equal parts with clear counts, and marks × 30° yields the angle at once, quick and accurate. Another example: the hour hand moving from 2 to 5 walks 3 marks, turning 3 × 30° = 90°. A common mistake is to think each mark is 60°, mixing 'minutes' with 'marks'; the 12 marks carry 360°, so a mark is 30°, while one minute of time moves the minute hand one small step of 6° — the two systems must not be confused.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• A full circle totals 360°
• Sector angle = fraction × 360°
• 14 = 90°, 12 = 180°, 25% = 90°
• Fraction from angle: 120° ÷ 360° = 13
• Each clock mark is 30°