Explore the concepts of calculating areas related to circles, including sectors, segments, and combinations with other geometric shapes.
Areas Related to Circles - Quick Look Revision Guide
Your 1-page summary of the most exam-relevant takeaways from Mathematics.
This compact guide covers 20 must-know concepts from Areas Related to Circles aligned with Class X preparation for Mathematics. Ideal for last-minute revision or daily review.
Complete study summary
Essential formulas, key terms, and important concepts for quick reference and revision.
Key Points
Define sector and segment of a circle.
A sector is the part of a circle enclosed by two radii and an arc. A segment is the part between a chord and its arc. Example: A pizza slice is a sector.
Formula for area of a sector.
Area = (θ/360) × πr², where θ is the angle in degrees and r is the radius. Example: For θ=30° and r=4cm, area ≈ 4.19cm².
Length of an arc formula.
Length = (θ/360) × 2πr. It's part of the circumference. Example: For θ=60° and r=21cm, length = 22cm.
Area of a segment calculation.
Area of segment = Area of sector - Area of triangle. Example: For r=21cm and θ=120°, area ≈ 462 - 441√3/4 cm².
Difference between minor and major sectors.
Minor sector has a smaller angle (θ), major sector has (360°-θ). Example: θ=30° means major sector angle is 330°.
Quadrant area and circumference relation.
A quadrant is a 90° sector. Area = (1/4)πr². If circumference is 22cm, radius is 3.5cm, area ≈ 9.625cm².
Clock hand area sweep.
Minute hand sweeps θ=30° in 5 mins. For length=14cm, area swept ≈ 25.67cm².
Chord subtending right angle area.
For r=10cm and θ=90°, minor segment area = (πr²/4) - (r²/2) ≈ 28.5cm².
Horse grazing area problem.
Horse tied with 5m rope in a corner grazes a 90° sector. Area = (90/360)π(5)² ≈ 19.625m².
Brooch wire length and sector area.
Diameter=35mm, radius=17.5mm. Total wire = circumference + 5 diameters ≈ 285mm. Each sector area ≈ 96.25mm².
Umbrella ribs area between.
8 ribs divide circle into 8 equal sectors. For r=45cm, area between two ribs ≈ 795.5cm².
Wiper blade cleaning area.
Each wiper sweeps 115° with 25cm blade. Total area ≈ 2 × (115/360)π(25)² ≈ 1254.8cm².
Lighthouse warning area.
Sector angle=80°, radius=16.5km. Area = (80/360)π(16.5)² ≈ 189.97km².
Round table cover design cost.
6 designs on r=28cm cover. Each design area ≈ (1/6)π(28)² - area of equilateral triangle. Cost ≈ ₹0.35 × total area.
Correct formula for sector area.
Area = (θ/360)πR². Option D is correct if θ is in degrees and R is radius.
Major segment area calculation.
Major segment area = πr² - minor segment area. Example: For r=15cm and θ=60°, major segment ≈ 3.14×225 - 20.4375 ≈ 686.0625cm².
Sector angle to arc length relation.
Arc length increases linearly with angle θ. For fixed r, double θ means double arc length.
Misconception: Segment area includes triangle.
No, segment area is sector area minus triangle area. Remember to subtract the triangle.
Real-world application: Fan blades.
Fan blades sweep a sector area. Calculating area helps in design and material estimation.
Memory hack: Sector like pizza slice.
Visualize sector as a pizza slice: angle is how wide you cut, radius is size of pizza.
Explore the properties, types, and theorems related to triangles, including congruence and similarity, to solve geometric problems effectively.
Coordinate Geometry explores the relationship between algebra and geometry through the use of coordinate systems to represent geometric shapes and solve problems.
Explore the basics of trigonometry, including angles, triangles, and the fundamental trigonometric ratios: sine, cosine, and tangent.
Explore real-world applications of trigonometry in measuring heights, distances, and angles in various fields such as astronomy, navigation, and architecture.
Explore the properties, theorems, and applications of circles in geometry, including tangents, chords, and angles subtended by arcs.
