Class 10 / Maths
Class 10 Maths Areas Related to Circles Notes
Notes on circumference, area of circle, sector, segment, and shaded region questions.
Topic introduction
Areas Related to Circles is a formula and visual-breakdown chapter. Before calculating, identify whether the question uses a full circle, semicircle, quadrant, sector, ring, or shaded region made from combined shapes.
Key points
- Circumference of a circle is 2 pi r and area is pi r^2.
- Area of a sector with angle theta is theta/360 x pi r^2.
- Length of an arc with angle theta is theta/360 x 2 pi r.
- Shaded region questions usually require subtracting smaller known areas from larger known areas.
Definitions
Sector: A region enclosed by two radii and the included arc.
Arc: A part of the circumference of a circle.
Segment: A region enclosed by a chord and the corresponding arc.
Quadrant: One-fourth of a circle.
Formulas
- Circumference = 2 pi r
- Area of circle = pi r^2
- Area of sector = theta/360 x pi r^2
- Arc length = theta/360 x 2 pi r
- Area of ring = pi(R^2 - r^2)
Examples
- Area of a 90 degree sector is one-fourth of the circle area.
- A ring with radii 14 cm and 7 cm has area pi(196 - 49).
- Area outside an inscribed circle in a square equals area of square minus area of circle.
Sector and arc
A sector is a part of a circle enclosed by two radii and an arc. Use the angle at the center to calculate sector area and arc length.
Keep the same unit throughout the calculation and round only at the end.
Shaded regions
Break the figure into standard shapes: circles, semicircles, quadrants, rectangles, triangles, and sectors.
Find the required region by adding and subtracting areas carefully.
Common mistakes
- Using diameter as radius without dividing by 2.
- Forgetting the theta/360 factor in sector and arc questions.
- Mixing area units and length units.
- Subtracting the wrong region in shaded-area problems.
- Rounding early and carrying inaccurate values.
Quick practice
- Find the area of a circle of radius 7 cm.
- Find the circumference of a circle of radius 14 cm.
- Find the area of a 90 degree sector of radius 8 cm.
- Find the length of a 60 degree arc of radius 21 cm.
Related practice
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