Sector area

1Summary 重點摘要

Sector = slice of circle.

2Vocabulary cards Vocabulary

3Step by step Steps

1. Step 1: Know the sectorA slice of pizza is a sector: two radii set out from the centre with an arc held between them. Two conditions set the sector's size: the radius and the central angle, which is the angle between the two radii. A larger central angle makes a larger sector, and a longer radius makes a larger sector too. Why these two? Because a sector is a part of a circle: the share of the part is decided by the angle, and the size of the base is decided by the radius; neither can be missing. A common misunderstanding is watching the angle alone, as if all 90° sectors were equal, yet the 90° sectors of radius 2 and radius 5 differ greatly in area.
2. Step 2: The formulaSector area = whole circle area × central angle ÷ 360. Whatever fraction the central angle is of 360°, the sector's area is that same fraction of the whole circle, and the two keep perfect step. Why in step? Because a sector is cut from the circle in proportion to its angle, so the angle's fraction is the area's fraction. The thinking is direct: find the whole circle first, then take the sector's part of it; for example 90° is a quarter of 360°, so the sector is a quarter of the circle. A common error is skipping the whole circle and multiplying the angle by the radius, which gives a meaningless number; the whole circle always comes first.
3. Step 3: The quarter circleFor a sector of 90°: 90 ÷ 360 = 14, so its area is a quarter of the circle. With radius 2 cm the whole circle is 3.14 × 2 × 2 = 12.56 cm², and the sector is 12.56 ÷ 4 = 3.14 cm². This is the basic example of sector area, and quarter circles are common in daily life, such as a pizza cut into four equal slices. The order is whole circle first, share second. A common error is writing 360 ÷ 90 instead of 90 ÷ 360, giving 4 times instead of a quarter, the complete reverse; afterwards, check that the sector's area is smaller than the whole circle's, since anything larger must be wrong.
4. Step 4: The semicircleA sector of 180° is a semicircle: 180 ÷ 360 = 12, half of the circle. With radius 2 cm the whole circle is 12.56 cm², so the semicircle is 12.56 ÷ 2 = 6.28 cm². The semicircle is an important special case, because 180° is exactly half of 360°, and the diameter serves as its straight edge. Remembering this case helps verify general results: any sector's area must lie between 0 and the whole circle's area, with the semicircle right in the middle. A common error is mixing up the semicircle with the quarter circle, swapping 12 and 14; a quick comparison with the visually expected size tells them apart.
5. Step 5: Any angleFor a sector with central angle 60° and radius 3 cm: the whole circle is 3.14 × 3 × 3 = 28.26 cm²; the share is 60 ÷ 360 = 16; and the sector is 28.26 ÷ 6 = 4.71 cm². Three tidy steps: whole circle, share, sector, each written clearly so checking is easy. Why is the share 16? Because 60° is one sixth of 360°, and six such sectors would fit together into a full circle. A common error is computing 3 × 3 as 6 instead of 9, which makes the whole circle 18.84 and the sector wrong as well; a square is a self-multiplication and demands extra care.
6. Step 6: Back to the whole circleFrom the sector's area the whole circle can be recovered: a 90° sector of 3.14 cm² is a quarter of its circle, so the whole circle = 3.14 × 4 = 12.56 cm². Why does the reversal work? Because multiplying and dividing undo each other: forwards the circle is divided by 4, and backwards it is multiplied by 4. This reverse thinking appears often in word problems, such as finding the radius from a sector's area and angle. Both directions deserve practice, so the formula comes alive. A common error is dividing again when reversing, producing a smaller number like 0.785; a sector is only a part, and going from the part to the whole always multiplies.

4Diagrams Diagrams

5Worked example Example

6Common mistakes Mistakes

7Quick check Quick check

8Recap Recap

• A sector is a part of a circle (a pizza slice)
• Sector area = circle area × angle ÷ 360
• 90° = quarter circle; 180° = semicircle
• Find the whole circle first, then scale down
• Multiply back from a sector to get the whole circle