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Areas Related to Circles

Areas Related to Circles

Action PlanNotesQuestionsMock-Test
Study Action Plan (Sorted by Difficulty)
Real-Life Applications (Sectors)
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Sector and Segment Relations
Accuracy: 0%
Total Attempted0
Total Correct0
Total Incorrect0
Difficulty Level:
High
Length of an Arc
Accuracy: 14%
Total Attempted7
Total Correct1
Total Incorrect6
Difficulty Level:
High
Area of a Sector
Accuracy: 29%
Total Attempted7
Total Correct2
Total Incorrect5
Difficulty Level:
High
Major vs Minor Calculations
Accuracy: 40%
Total Attempted15
Total Correct6
Total Incorrect9
Difficulty Level:
High
Trigonometric Application in Circle Geometry
Accuracy: 69%
Total Attempted13
Total Correct9
Total Incorrect4
Difficulty Level:
Medium
Area of a Segment
Accuracy: 83%
Total Attempted29
Total Correct24
Total Incorrect5
Difficulty Level:
Low

Learning Objectives

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Learning Objectives

  • Understand the derivation and formula for calculating the area of a sector of a circle with radius rr and central angle Θ\Theta in degrees.
  • Calculate the length of an arc of a sector of a circle with radius rr and central angle Θ\Theta in degrees using the appropriate formula.
  • Determine the area of a segment of a circle by subtracting the area of the triangle formed by the chord from the area of the sector.
  • Apply trigonometric identities and properties, such as cosine and sine rules, to solve problems related to segments and sectors of circles.
  • Analyze the relationships between major and minor sectors and segments, including complementary angles and areas.

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Revision Notes & Summary

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Chapter Notes

Area of a Sector

  • Definition: A sector of a circle is the region enclosed by two radii and the corresponding arc.
  • Formula Derivation:
    • The area of a full circle is πr2\pi r^2.
    • For a sector with central angle Θ\Theta (in degrees), the area is proportional to the angle.
    • Formula: Area of the sector=Θ360×πr2\text{Area of the sector} = \frac{\Theta}{360} \times \pi r^2

Length of an Arc

  • Definition: The arc length is the distance along the curved line forming the arc.
  • Formula Derivation:
    • The circumference of a full circle is 2πr2\pi r.
    • For an arc with central angle Θ\Theta, the length is proportional to the angle.
    • Formula: Length of the arc=Θ360×2πr\text{Length of the arc} = \frac{\Theta}{360} \times 2\pi r

Area of a Segment

  • Definition: A segment of a circle is the region enclosed by a chord and the corresponding arc.
  • Calculation:
    • Area of the segment = Area of the sector - Area of the triangle formed by the chord.
    • Formula: Area of the segment=Θ360×πr2−Area of △OAB\text{Area of the segment} = \frac{\Theta}{360} \times \pi r^2 - \text{Area of } \triangle OAB

Trigonometric Application in Circle Geometry

  • Use of Trigonometric Identities:
    • Cosine Rule: Used to find the length of a chord or solve for angles.
    • Sine Rule: Used to solve for unknown sides or angles in the triangle formed by the radii and the chord.

Sector and Segment Relations

  • Major and Minor Sectors:
    • The major sector is the larger area, while the minor sector is the smaller.
    • Relation: Area of major sector = πr2\pi r^2 - Area of minor sector.
  • Major and Minor Segments:
    • The major segment is the larger area, while the minor segment is the smaller.
    • Relation: Area of major segment = πr2\pi r^2 - Area of minor segment.

Examples

  • Example 1: Calculate the area of a sector with radius 4 cm and angle 30°.
    • Solution: Area=30360×π×42=4.19 cm2\text{Area} = \frac{30}{360} \times \pi \times 4^2 = 4.19 \text{ cm}^2
  • Example 2: Find the area of a segment with radius 21 cm and angle 120°.
    • Solution: Use the formula for the area of a segment and apply trigonometric identities to find the area of the triangle.

Summary

  • Arc Length: Length of arc=Θ360×2πr\text{Length of arc} = \frac{\Theta}{360} \times 2\pi r
  • Sector Area: Area of sector=Θ360×πr2\text{Area of sector} = \frac{\Theta}{360} \times \pi r^2
  • Segment Area: Area of segment=Area of sector−Area of triangle\text{Area of segment} = \text{Area of sector} - \text{Area of triangle}
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Practice Test – MCQs, True/False

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Multiple Choice Questions

A.

60∘60^\circ

B.

90∘90^\circ

C.

120∘120^\circ

D.

180∘180^\circ
Correct Answer: D

Solution:

Using the formula Θ360×πr2=78.5\frac{\Theta}{360} \times \pi r^2 = 78.5, solving for Θ\Theta gives Θ=180∘\Theta = 180^\circ.

Chapter Concept:

Area of a Sector

A.

30°

B.

60°

C.

45°

D.

90°
Correct Answer: B

Solution:

The arc length of a sector is given by Θ360×2πr\frac{\Theta}{360} \times 2\pi r. Solving Θ360×2×3.14×8=8.38\frac{\Theta}{360} \times 2 \times 3.14 \times 8 = 8.38 gives Θ=60°\Theta = 60°.

Chapter Concept:

Area of a Sector

A.

10 cm

B.

12 cm

C.

14 cm

D.

16 cm
Correct Answer: A

Solution:

Using the formula for the area of a sector, θ360×πr2=50\frac{\theta}{360} \times \pi r^2 = 50, we substitute θ=60\theta = 60 and solve for rr: 60360×πr2=50⇒16×πr2=50⇒πr2=300⇒r2=300π⇒r≈10\frac{60}{360} \times \pi r^2 = 50 \Rightarrow \frac{1}{6} \times \pi r^2 = 50 \Rightarrow \pi r^2 = 300 \Rightarrow r^2 = \frac{300}{\pi} \Rightarrow r \approx 10 cm.

Chapter Concept:

Major vs Minor Calculations

A.

78.5 cm²

B.

21.5 cm²

C.

50 cm²

D.

28.5 cm²
Correct Answer: B

Solution:

The area of the segment is the area of the sector minus the area of the triangle. The sector area is 90360×π×102=78.5\frac{90}{360} \times \pi \times 10^2 = 78.5 cm². The triangle area is 12×10×10×sin⁡90∘=50\frac{1}{2} \times 10 \times 10 \times \sin 90^\circ = 50 cm². Thus, the segment area is 78.5−50=28.578.5 - 50 = 28.5 cm².

Chapter Concept:

Area of a Segment

A.

14.66 cm

B.

8.8 cm

C.

14.66 cm

D.

16.8 cm
Correct Answer: D

Solution:

The length of the arc is given by θ360×2πr=120360×2×π×7=14.66\frac{\theta}{360} \times 2\pi r = \frac{120}{360} \times 2 \times \pi \times 7 = 14.66 cm.

Chapter Concept:

Area of a Segment

A.

45°

B.

60°

C.

90°

D.

120°
Correct Answer: B

Solution:

The area of a sector is given by the formula: Area of the sector = θ360×πr2\frac{\theta}{360} \times \pi r^2. Rearranging to find θ\theta, we have θ=360×Area of the sectorπr2\theta = \frac{360 \times \text{Area of the sector}}{\pi r^2}. Substituting the given values, θ=360×38.53.14×72=60°\theta = \frac{360 \times 38.5}{3.14 \times 7^2} = 60°.

Chapter Concept:

Trigonometric Application in Circle Geometry

A.

21.5 cm²

B.

23.6 cm²

C.

24.1 cm²

D.

25.7 cm²
Correct Answer: B

Solution:

The area of the sector OAPB is calculated as 60360×π×102=52.33 cm2\frac{60}{360} \times \pi \times 10^2 = 52.33 \text{ cm}^2. The area of triangle OAB can be calculated using the formula 12×r2×sin⁡(θ)\frac{1}{2} \times r^2 \times \sin(\theta), which gives 12×102×sin⁡(60∘)=43.25 cm2\frac{1}{2} \times 10^2 \times \sin(60^\circ) = 43.25 \text{ cm}^2. Therefore, the area of the segment APB is 52.33−43.25=9.08 cm252.33 - 43.25 = 9.08 \text{ cm}^2.

Chapter Concept:

Area of a Segment

A.

35.4 cm²

B.

37.2 cm²

C.

38.6 cm²

D.

40.1 cm²
Correct Answer: D

Solution:

The area of the sector OAPB is 150360×π×92=106.05 cm2\frac{150}{360} \times \pi \times 9^2 = 106.05 \text{ cm}^2. The area of triangle OAB is 12×92×sin⁡(150∘)=40.5 cm2\frac{1}{2} \times 9^2 \times \sin(150^\circ) = 40.5 \text{ cm}^2. Therefore, the area of the segment APB is 106.05−40.5=65.55 cm2106.05 - 40.5 = 65.55 \text{ cm}^2.

Chapter Concept:

Area of a Segment

A.

π4\frac{\pi}{4} cm

B.

5π9\frac{5\pi}{9} cm

C.

5π4\frac{5\pi}{4} cm

D.

π2\frac{\pi}{2} cm
Correct Answer: C

Solution:

The length of the arc is given by the formula Θ360×2πr\frac{\Theta}{360} \times 2\pi r. Substituting the values, we get 45360×2π×10=5π4\frac{45}{360} \times 2\pi \times 10 = \frac{5\pi}{4} cm.

Chapter Concept:

Length of an Arc

A.

7π7\pi cm

B.

14π14\pi cm

C.

7π2\frac{7\pi}{2} cm

D.

14π3\frac{14\pi}{3} cm
Correct Answer: A

Solution:

The length of the arc is calculated using Θ360×2πr\frac{\Theta}{360} \times 2\pi r. Substituting the given values, we have 90360×2π×14=7π\frac{90}{360} \times 2\pi \times 14 = 7\pi cm.

Chapter Concept:

Length of an Arc

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Areas Related to Circles

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True or False

Correct Answer: True

Solution:

The area of the major sector is calculated as the total area of the circle minus the area of the minor sector, as stated in the text.

Chapter Concept :

Sector and Segment Relations

Correct Answer: True

Solution:

Since the length of the arc is directly proportional to the central angle Θ\Theta (i.e., Θ360×2πr\frac{\Theta}{360} \times 2\pi r), doubling Θ\Theta results in doubling the arc length.

Chapter Concept :

Length of an Arc

Correct Answer: True

Solution:

The formula for the area of a sector with angle θ\theta is correctly given by θ360×πr2\frac{\theta}{360} \times \pi r^2, as per the excerpt.

Chapter Concept :

Area of a Segment

Correct Answer: True

Solution:

The area of a segment is indeed found by subtracting the area of the triangle formed by the chord from the area of the sector.

Chapter Concept :

Area of a Segment

Correct Answer: True

Solution:

The formula for the area of a sector is derived by considering the proportion of the angle Θ\Theta to the full circle (360 degrees), thus giving Θ360×πr2\frac{\Theta}{360} \times \pi r^2.

Chapter Concept :

Area of a Sector

Correct Answer: True

Solution:

The length of the arc is found by taking the fraction Θ360\frac{\Theta}{360} of the circumference of the circle, which is 2πr2\pi r.

Chapter Concept :

Area of a Sector

Correct Answer: True

Solution:

The formula for the area of a sector with angle θ\theta in degrees and radius rr is θ360×πr2\frac{\theta}{360} \times \pi r^2, as derived using the unitary method.

Chapter Concept :

Real-Life Applications (Sectors)

Correct Answer: True

Solution:

The area of the major sector is indeed the total area of the circle minus the area of the minor sector, as stated in the excerpts.

Chapter Concept :

Major vs Minor Calculations

Correct Answer: True

Solution:

The area of the major sector is calculated by subtracting the area of the minor sector from the total area of the circle, as stated in the excerpt.

Chapter Concept :

Sector and Segment Relations

Correct Answer: False

Solution:

The area of a minor segment is calculated by subtracting the area of the corresponding triangle from the area of the minor sector, not adding.

Chapter Concept :

Major vs Minor Calculations

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