Areas Related to Circles · Class 10 Mathematics: 44 flashcards
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1. What is a sector?
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The region between two radii of a circle and the arc joining their ends.
2. Which is the minor sector?
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The smaller of the two pieces cut off by the two radii.
3. What fraction of the circle is a sector of angle θ?
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$\frac{\theta}{360}$.
4. Give the formula for the area of a sector.
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$\frac{\theta}{360} \times \pi r^2$.
5. Give the formula for the length of an arc.
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$\frac{\theta}{360} \times 2\pi r$.
6. Give the formula for the perimeter of a sector.
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arc $+ 2r$, that is $\frac{\theta}{360} \times 2\pi r + 2r$.
7. What is a quadrant?
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A sector of 90°, a quarter of the circle.
8. Find the area of a 60° sector of a circle of radius 21 cm, with π = 22/7.
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$\frac16 \times \frac{22}{7} \times 441 = 231$ cm².
9. Find the arc length of that sector.
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$\frac16 \times 2 \times \frac{22}{7} \times 21 = 22$ cm.
10. Find its perimeter.
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$22 + 2 \times 21 = 64$ cm.
11. Find the area of a quadrant of radius 14 cm.
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$\frac14 \times \frac{22}{7} \times 196 = 154$ cm².
12. Why do textbook radii tend to be 7, 14, 21 or 28?
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Because $\frac{22}{7}$ then cancels and the answers come out as whole numbers.
13. What should you do if the radius is not a multiple of 7?
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Leave the answer in terms of π, or use π ≈ 3.14.
14. A sector of a circle of radius 6 cm has area 3π cm². Find its angle.
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$\theta = \frac{360 \times 3\pi}{36\pi} = 30°$.
15. How are the minor and major sector areas related?
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They add to the whole area, $\pi r^2$, because the angles add to 360°.
16. What check confirms a sector formula is right?
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Set θ = 360; it should reduce to the whole circle's area or circumference.
17. In what units is a sector's area given?
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Square units — cm², m².
18. In what units is an arc length given?
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Ordinary length units — cm, m.
19. A semicircle is a sector of what angle?
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180°.
20. The arc of a quadrant of radius $r$ is?
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$\frac{\pi r}{2}$ — a quarter of $2\pi r$.
21. What happens to a sector's area if the angle doubles?
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It doubles, since the area is proportional to θ.
22. What happens to a sector's area if the radius doubles?
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It becomes four times as large, since the area depends on $r^2$.
23. What is a segment of a circle?
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The region between a chord and one of the two arcs it cuts off.
24. How does a segment differ from a sector?
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A sector is bounded by two radii and an arc; a segment by one chord and an arc.
25. Give the formula for the area of a minor segment.
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Area of the sector minus the area of the triangle formed by the two radii and the chord.
26. Why is triangle OAB always isosceles?
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Because OA and OB are both radii of the same circle.
27. Give the formula for the area of triangle OAB.
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$\frac12 r^2 \sin\theta$.
28. What is special about a central angle of 60°?
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The triangle is equilateral, so the chord equals the radius and the area is $\frac{\sqrt3}{4}r^2$.
29. What is the triangle's area when θ = 90°?
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$\frac{r^2}{2}$ — a right triangle with both legs equal to r.
30. What is the triangle's area when θ = 120°?
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$\frac{\sqrt3}{4}r^2$, the same as at 60°, because $\sin 120° = \sin 60°$.
31. Which central angles does the syllabus allow for segment problems?
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60°, 90° and 120° only.
32. A chord of a circle of radius 10 cm subtends 90° at the centre. Find the minor segment's area.
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$25\pi - 50 \approx 28.5$ cm².
33. A chord of a circle of radius 21 cm subtends 60°. Find the minor segment's area, with π = 22/7.
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$231 - \frac{441\sqrt3}{4} \approx 40.05$ cm².
34. How do you find the major segment's area?
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Subtract the minor segment from the whole circle's area, $\pi r^2$.
35. Give the formula for the length of a chord subtending θ.
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$2r\sin\frac{\theta}{2}$.
36. How long is the chord when θ = 60°?
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Exactly r — the chord equals the radius.
37. How long is the chord when θ = 90°?
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$r\sqrt2$.
38. How long is the chord when θ = 120°?
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$r\sqrt3$.
39. Give the perimeter of a segment.
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Arc plus chord — no radii.
40. How does that differ from a sector's perimeter?
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A sector's perimeter is arc plus 2r; a segment's is arc plus the chord.
41. Why should $\sqrt3$ be kept exact until the end?
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A segment is a small difference between two large numbers, so early rounding distorts it badly.
42. What does a correct segment answer usually look like?
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A π term minus a surd or an integer.
43. What error does an answer with no subtraction in it suggest?
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That the triangle was never taken away — the sector's area was given instead.
44. From what should the triangle be subtracted?
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From the sector containing it, not from the whole circle.
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