Areas Related To CirclesClass 10 Mathematics Important Points

A sector is the portion of a circular region enclosed by two radii and the corresponding arc. The smaller area is the minor sector and the larger area is the major sector.

A segment is the portion of a circular region enclosed between a chord and the corresponding arc. The smaller area is the minor segment, and the larger one is the major segment.

The area of a sector with angle $\theta$ (in degrees) and radius $r$ is given by the formula: Area $= \frac{\theta}{360} \times \pi r^2$. The area is proportional to the central angle.

The length of an arc of a sector with angle $\theta$ (in degrees) and radius $r$ is calculated as: Arc Length $= \frac{\theta}{360} \times 2 \pi r$. This is a fraction of the total circumference.

The area of a segment is found by subtracting the area of the corresponding triangle from the area of the sector. Area of Segment = Area of Sector OAPB - Area of $\triangle$ OAB.

To find the area of the triangle (e.g., $\triangle$ OAB) formed by the two radii and the chord, you can use trigonometry or geometric properties. For an angle $\theta$ and radius $r$, the area is $\frac{1}{2} r^2 \sin \theta$.

When the central angle of a sector is $60^\circ$, the triangle formed by the two radii and the chord is an equilateral triangle. Its area is $\frac{\sqrt{3}}{4} r^2$, where $r$ is the radius.

When the central angle is $90^\circ$ (a quadrant), the triangle formed is a right-angled isosceles triangle. Its area is $\frac{1}{2} \times \text{base} \times \text{height} = \frac{1}{2} r^2$.

The area of a major sector can be found in two ways. Either subtract the minor sector's area from the total circle's area: $\pi r^2 - \text{Area of Minor Sector}$, or use the reflex angle $(360^\circ - \theta)$ in the sector formula.

The area of a major segment is calculated by subtracting the area of the minor segment from the total area of the circle. Area of Major Segment $= \pi r^2 - \text{Area of Minor Segment}$.

A quadrant of a circle is a sector with a central angle of $90^\circ$. Its area is exactly one-fourth of the total area of the circle, given by the formula Area $= \frac{1}{4} \pi r^2$.

The area swept by a minute hand in a given time is the area of a sector. The length of the hand is the radius $r$. The angle is found by knowing the minute hand moves $360^\circ$ in 60 minutes, which is $6^\circ$ per minute.

The formula for the area of a sector, $\frac{\theta}{360} \times \pi r^2$, can sometimes be written as $\frac{\theta}{720} \times 2 \pi r^2$. Both forms are equivalent and give the same result.