A circular sector, relevant to class 10 geometry, is the region of a circle enclosed by two radii and the arc they subtend. Think of it as a slice of pizza or pie.
Essentially, in the context of Class 10 mathematics (particularly geometry), understanding circular sectors involves learning about their properties, calculating their areas, and finding the lengths of their corresponding arcs. This builds upon the fundamental knowledge of circles, radii, diameters, circumference, and area.
Key Concepts Related to Circular Sectors in Class 10:
- Definition: A circular sector is a portion of a circle bounded by two radii and the intercepted arc.
- Central Angle (θ): The angle formed by the two radii at the center of the circle, measured in degrees.
- Arc Length (l): The length of the arc that forms the curved boundary of the sector.
- Area of the Sector (A): The region enclosed by the two radii and the arc.
- Minor Sector: The sector with a central angle less than 180 degrees.
- Major Sector: The sector with a central angle greater than 180 degrees. The major sector includes everything else besides the minor sector.
Formulas for Class 10:
| Property | Formula | Description |
|---|---|---|
| Arc Length (l) | l = (θ/360) * 2πr | θ is the central angle in degrees, r is the radius of the circle. |
| Area (A) | A = (θ/360) * πr2 | θ is the central angle in degrees, r is the radius of the circle. |
| Area (A) | A = (1/2) l r | l is the arc length, r is the radius of the circle. |
Example Problem:
Question: A circle has a radius of 7 cm. A sector of the circle has a central angle of 90 degrees. Find the area of the sector and the length of its arc.
Solution:
-
Area of the sector:
- A = (θ/360) * πr2
- A = (90/360) π (7)2
- A = (1/4) π 49
- A = (49π)/4 cm2 (Approximately 38.48 cm2)
-
Arc Length:
- l = (θ/360) * 2πr
- l = (90/360) 2 π * 7
- l = (1/4) * 14π
- l = (7π)/2 cm (Approximately 10.99 cm)
Importance in Class 10:
Understanding circular sectors is vital for:
- Solving problems related to areas of combined figures (e.g., a square with a sector cut out).
- Applying geometrical concepts to real-world scenarios (e.g., calculating the area covered by a sprinkler).
- Building a strong foundation for more advanced topics in trigonometry and calculus.
In summary, a circular sector in Class 10 mathematics is a fundamental geometrical shape defined by two radii and an arc of a circle. Its area and arc length are key calculations learned at this level, with applications in problem-solving and real-world scenarios.
