Areas Related to Circles — RBSE Class 10 (Mathematics)

A pizza slice, a clock's swept minute hand, the shaded corner of a design — all are parts of a circle. Once you can find the area of a whole circle, the trick is simple: a sector is just the fraction $\frac{\theta }{360}$ of the whole. This chapter is pure mensuration — draw carefully, add and subtract areas, and the marks follow.

1. Circumference and area of a full circle

For radius $r$: $\text{Circumference}=2\pi r,\text{Area}=\pi r^2$

Use $\pi =\frac{22}{7}$ when the radius is a multiple of 7, otherwise $\pi =3.14$ (follow what the question specifies).

2. Sector — the pie slice

A sector is bounded by two radii and an arc; the angle between the radii is $\theta$. Since the slice is the fraction $\frac{\theta }{360}$ of the whole circle:

$\text{Length of arc}=\frac{\theta }{360}\times 2\pi r$ $\text{Area of sector}=\frac{\theta }{360}\times \pi r^2$

A handy alternative: Area of sector $=\frac{1}{2}\times (\text{arc length})\times r$.

3. Segment — the slice minus the triangle

A chord cuts a circle into two segments. The minor segment is the sector minus the triangle formed by the two radii:

$\text{Area of minor segment}=\frac{\theta }{360}\pi r^2-\frac{1}{2}{r}^2\sin \theta$

(The triangle's area is $\frac{1}{2}{r}^2\sin \theta$; for the special angles 60°/90°/120° you can also compute it directly.) The major segment = area of circle − minor segment.

4. Combined and shaded figures

The real exam questions layer shapes: a circle inside a square, four quadrants at the corners of a square, a running track (two semicircles + a rectangle), a design of overlapping arcs. The method never changes:

Shaded area = (area of the outer/whole figure) − (areas removed) [+ areas added].

Break the figure into circles, sectors, triangles, squares and rectangles whose areas you know, then add and subtract.

5. Two useful facts

• Angle swept by a clock hand: the minute hand sweeps $6°$ per minute (360° in 60 min); the hour hand $0.5°$ per minute. Convert time to θ, then use the sector formulas.
• Areas of two circles combine by their squares: if radii are in ratio $a:b$, areas are in ratio $a^2:b^2$ (same idea as similar figures).

6. Worked idea

Find the area of a sector of radius 21 cm with angle 60° (take $\pi =\frac{22}{7}$). $\frac{60}{360}\times \frac{22}{7}\times 21^2=\frac{1}{6}\times \frac{22}{7}\times 441=\frac{1}{6}\times 1386=231{\text{ cm}}^2.$

7. Closing thought

Two formulas scaled by $\frac{\theta }{360}$ do most of the work; segments just subtract a triangle. The exam premium is on reading the figure and deciding what to add or subtract. Sketch each component, write its area on the diagram, and keep $\pi$ consistent — the RBSE board sets at least one sector/segment or shaded-region problem every year.